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Sequence

In mathematics, a sequence is an ordered list of elements, typically numbers, arranged according to a specific rule or pattern, which may be finite or infinite in length.^[1] Formally, a sequence can be defined as a function from the natural numbers (or a subset thereof, such as the positive integers starting from 1 or 0) to a codomain like the real numbers, where each term is denoted as a_n for the nth position.^[2] Sequences are fundamental in mathematical analysis, serving as the building blocks for concepts like limits, convergence, and infinite series, which underpin differential equations, approximations, and many areas of advanced mathematics. Common types of sequences include arithmetic sequences, where each term after the first is obtained by adding a constant difference d to the preceding term (e.g., 2, 5, 8, 11, ... with d = 3), and geometric sequences, where each term is multiplied by a constant ratio r (e.g., 3, 6, 12, 24, ... with r = 2).^[3] Other notable sequences encompass the Fibonacci sequence, defined recursively by F_1 = 1, F_2 = 1, and F_n = F_{n-1} + F_{n-2} for n > 2 (yielding 1, 1, 2, 3, 5, 8, ...), which appears in natural patterns like plant growth and is influential in number theory and computer science.^[4] Sequences can also be finite, terminating after a fixed number of terms, or infinite, continuing indefinitely, with the latter often studied for properties like monotonicity (always increasing or decreasing) or boundedness (confined within limits).^[5] A key aspect of infinite sequences is convergence: a sequence \{a_n\} converges to a limit L if, for every \epsilon > 0, there exists an N such that for all n > N, |a_n - L| < \epsilon, enabling the analysis of behavior as n approaches infinity.^[6] This property is crucial for defining continuity in functions and sums of series, where the partial sums form a sequence themselves.^[7] Beyond pure mathematics, sequences model real-world phenomena, such as population growth in arithmetic or geometric progressions, financial compounding, and algorithmic iterations in computing.^[8]

Basic Concepts and Examples

Intuitive Examples

A sequence in mathematics can be intuitively understood as an ordered list of numbers, where each number is associated with a specific position, often indexed by positive integers starting from 1 or 0.^[9] This ordering distinguishes sequences from mere sets, as the position of each term matters; for example, rearranging the terms would yield a different sequence.^[1] One of the simplest examples is the sequence of natural numbers: $1, 2, 3, 4, \dots , which continues indefinitely and represents the basic counting process.^[10] Another straightforward infinite sequence is the even numbers: $2, 4, 6, 8, \dots , where each term is twice its position if indexed from 1 (a_n = 2n).^[9] Finite sequences also arise naturally, such as the first four odd numbers: $1, 3, 5, 7, which terminates after a fixed number of terms.^[10] Arithmetic sequences provide an intuitive progression with a constant difference between terms, like $2, 5, 8, 11, \dots (common difference of 3, given by a_n = 3n - 1), modeling steady increases such as daily temperature rises.^[9] Geometric sequences, in contrast, multiply by a constant ratio, as in $2, 4, 8, 16, \dots (ratio of 2), illustrating exponential growth like bacterial population doubling.^[10] The Fibonacci sequence offers a recursive intuitive example: starting with 1 and 1, each subsequent term is the sum of the previous two, yielding $1, 1, 2, 3, 5, 8, 13, \dots ; it models natural growth patterns, such as the idealized reproduction of rabbit pairs where each pair produces a new pair monthly after maturity.^[11] Alternating sequences, like $3, 5, 3, 5, \dots (given by a_n = 4 + (-1)^n), highlight oscillatory behavior, akin to flipping between two states.^[9]

Standard Notation and Indexing

In mathematics, sequences are formally defined as functions from an ordered index set, typically the natural numbers, to a codomain such as the real numbers \mathbb{R}. The standard notation employs subscripts to denote terms, with the general term written as a_n, where n serves as the index indicating the position. The full sequence is commonly expressed as (a_n)_{n=1}^\infty or \{a_n\}_{n=1}^\infty, signifying an infinite ordered list where the subscript n ranges over the positive integers starting from 1. This convention aligns with the sequence's role as a mapping a: \mathbb{N} \to \mathbb{R}, emphasizing order and potential repetition.^[12]^[13] Indexing typically begins at n=1 in pure mathematics to align with positive integers, though contexts like computer science or certain analytical treatments may start at n=0. For instance, the sequence of squares can be a_n = n^2 for n \geq 1, yielding $1, 4, 9, \dots, or adjusted to include $0 if indexing from 0. Finite sequences use bounded indices, denoted as (a_n)_{n=1}^k or listed explicitly as (a_1, a_2, \dots, a_k), such as (2, 3, 5, 7) for the first four primes. The subscript notation facilitates compact expressions for operations like limits, e.g., \lim_{n \to \infty} a_n = L.^[12]^[13]^[14] While function notation a(n) is occasionally used for clarity in definitions, subscript a_n is preferred for its brevity and tradition in analysis, avoiding confusion with general functions. Variations in starting index do not alter core properties like convergence, provided the domain is consistently specified; for example, shifting indices merely reindexes the terms without changing the tail behavior relevant to limits.^[13]^[12]

Recursive Definitions

A recursive definition of a sequence specifies one or more initial terms and a recurrence relation that expresses each subsequent term as a function of the preceding terms. This approach contrasts with explicit definitions, which provide a direct formula for the nth term without reference to prior terms. Recursive definitions are fundamental in discrete mathematics and are often used in conjunction with mathematical induction to prove properties of sequences.^[15] Formally, for a sequence (a_n)_{n=0}^\infty or (a_n)_{n=1}^\infty, the recursive definition consists of initial conditions, such as a_0 = c_0 or a_1 = c_1, and a recurrence relation a_n = f(a_{n-1}, a_{n-2}, \dots, a_{n-k}) for n > k, where k is the order of the recursion (the number of previous terms required). First-order recursions depend on a single prior term, while higher-order ones involve multiple. The order determines the number of initial conditions needed; for example, a second-order recursion requires two starting values. Such definitions enable iterative computation of terms but may require unfolding the recursion to find closed-form expressions for analysis.^[16]^[17]^[18] Arithmetic sequences provide a simple first-order example. Given an initial term a_1 and common difference d, the recursive formula is a_{n+1} = a_n + d for n \geq 1. For instance, starting with a_1 = 2 and d = 3, the sequence is 2, 5, 8, 11, ..., where each term adds 3 to the previous. This recursion directly reflects the constant difference property of arithmetic progressions.^[19]^[20] Geometric sequences are similarly defined recursively by a first-order relation. With initial term a_1 and common ratio r, the formula is a_{n+1} = r \cdot a_n for n \geq 1. For a_1 = 1 and r = 2, the sequence is 1, 2, 4, 8, ..., doubling each term. This captures the multiplicative growth inherent to geometric progressions.^[21]^[20] Higher-order recursions appear in more complex sequences, such as the Fibonacci sequence, a second-order linear recurrence. Defined by f_1 = 1, f_2 = 1, and f_n = f_{n-1} + f_{n-2} for n \geq 3, it generates 1, 1, 2, 3, 5, 8, 13, .... The factorial sequence offers a non-linear example: $0! = 1 and n! = n \cdot (n-1)! for n \geq 1, yielding 1, 1, 2, 6, 24, .... These examples illustrate how recursive definitions model combinatorial growth and are solvable via methods like generating functions or characteristic equations for linear cases.^[15]^[16]^[18]

Formal Properties

Mathematical Definition

In mathematics, a sequence is formally defined as a function whose domain is the set of natural numbers (typically starting from 1) and whose codomain is a specified set, such as the real numbers \mathbb{R}. More precisely, given a function f: \mathbb{N} \to X, where \mathbb{N} = \{1, 2, 3, \dots\} and X is any set (often \mathbb{R} or \mathbb{C}), the sequence is the ordered list of values f(1), f(2), f(3), \dots, denoted as \{a_n\}_{n=1}^\infty where a_n = f(n) for each n \in \mathbb{N}.^[1] This definition emphasizes the ordered nature of the elements, distinguishing sequences from unordered sets.^[22] The range of the sequence, consisting of the terms a_n, forms an infinite list that may or may not follow a discernible pattern, but the indexing by natural numbers ensures a well-defined order.^[23] For example, the sequence defined by f(n) = n yields \{1, 2, 3, \dots\}, while f(n) = \frac{1}{n} gives \left\{\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \dots\right\}. In some contexts, particularly in computer science or when including an initial term, the domain may start from 0, so \mathbb{N}_0 = \{0, 1, 2, \dots\}, but in real analysis, indexing from 1 is conventional to align with limit definitions. This functional perspective allows sequences to be analyzed using tools from set theory and topology, such as convergence, where the behavior of a_n as n \to \infty is studied.^[24] Sequences generalize to arbitrary index sets that are well-ordered, but the standard case uses countable indices for simplicity and applicability in calculus.^[25]

Finite and Infinite Sequences

A finite sequence is defined as an ordered list of elements where the domain of the indexing function is the set of the first n positive integers, \{1, 2, \dots, n\}, for some positive integer n. This results in a sequence with exactly n terms, denoted as (a_1, a_2, \dots, a_n) or \{a_k\}_{k=1}^n. For instance, the sequence defined by a_k = k^2 for k = 1 to $5 yields (1, 4, 9, 16, 25). Finite sequences are straightforward to compute in their entirety, including operations like summation, which is simply \sum_{k=1}^n a_k, without requiring limits.^[4]^[20]^[26] In contrast, an infinite sequence is an ordered list where the domain is the set of all positive integers, \{1, 2, 3, \dots\}, producing terms that continue indefinitely, denoted as (a_1, a_2, a_3, \dots) or \{a_n\}_{n=1}^\infty. An example is the harmonic sequence a_n = \frac{1}{n}, which begins (1, \frac{1}{2}, \frac{1}{3}, \dots). Infinite sequences form the foundation for studying convergence and limits, as their behavior is analyzed through the limit of terms as n approaches infinity, rather than a fixed endpoint. Summation of infinite sequences leads to infinite series, where the total sum, if it exists, is the limit of partial sums \lim_{n \to \infty} \sum_{k=1}^n a_k.^[4]^[1]^[20] The primary distinction between finite and infinite sequences lies in their length and analytical treatment: finite sequences terminate and allow complete enumeration, making them suitable for discrete computations, while infinite sequences require asymptotic analysis to determine properties like boundedness or monotonicity over unbounded indices. Both can be defined explicitly via a formula for the general term a_n or recursively, but infinite sequences often exhibit patterns that persist indefinitely, such as arithmetic progressions where a_{n+1} = a_n + d for constant d. The empty sequence, with n=0, is sometimes considered a finite sequence of length zero.^[4]^[1]^[20]

Monotonicity

In mathematics, a sequence \{a_n\} is said to be monotonic if it is either monotonically increasing or monotonically decreasing.^[27] A sequence is monotonically increasing if a_n \leq a_{n+1} for all n \in \mathbb{N}, and strictly increasing if the inequality is strict, i.e., a_n < a_{n+1}.^[28] Conversely, a sequence is monotonically decreasing if a_n \geq a_{n+1} for all n, and strictly decreasing if a_n > a_{n+1}.^[29] Monotonicity provides a fundamental property for analyzing sequence behavior, particularly in relation to convergence. The Monotone Convergence Theorem states that a monotonic sequence converges if and only if it is bounded.^[30] Specifically, if \{a_n\} is monotonically increasing and bounded above, it converges to its least upper bound (supremum); if monotonically decreasing and bounded below, it converges to its greatest lower bound (infimum).^[31] An unbounded monotonic sequence diverges to +\infty or -\infty, depending on the direction.^[32] For example, the sequence a_n = n is strictly increasing but unbounded above, hence diverges to +\infty.^[33] In contrast, a_n = 1 - \frac{1}{n} is strictly increasing and bounded above by 1, so it converges to 1.^[27] These properties are essential in real analysis, as they link order-preserving behavior to limit existence without requiring explicit computation.^[34]

Boundedness

In mathematics, a sequence \{a_n\}_{n=1}^\infty of real numbers is said to be bounded if there exist real numbers m and M such that m \leq a_n \leq M for all n \in \mathbb{N}.^[35] This condition ensures that all terms of the sequence lie within a finite interval [m, M].^[36] Equivalently, a sequence is bounded if the set \{a_n : n \in \mathbb{N}\} is bounded as a subset of \mathbb{R}.^[36] A sequence is bounded above if there exists a real number M such that a_n \leq M for all n \in \mathbb{N}; this M is an upper bound for the sequence.^[35] Similarly, it is bounded below if there exists a real number m such that a_n \geq m for all n \in \mathbb{N}, where m serves as a lower bound.^[36] A sequence is bounded if and only if it is both bounded above and bounded below.^[35] If no such finite bounds exist, the sequence is unbounded.^[35] For example, the sequence a_n = \frac{1}{n} is bounded, as $0 < a_n \leq 1 for all n \geq 1.^[35] In contrast, the sequence a_n = n is unbounded above (and hence unbounded), since its terms grow without limit.^[35] Another illustrative case is a_n = (-1)^n, which is bounded (with -1 \leq a_n \leq 1) but oscillates indefinitely.^[36] Boundedness is a fundamental property in the study of sequences, particularly in relation to convergence. Every convergent sequence is bounded: if \{a_n\} \to L, then for \epsilon = 1, there exists N such that |a_n - L| < 1 for all n > N, implying |a_n| \leq |L| + 1 for all n (and adjusting for the finite initial terms).^[36] However, the converse does not hold, as the bounded sequence a_n = (-1)^n diverges by oscillation.^[36] Unbounded sequences necessarily diverge, as they cannot approach any finite limit.^[35] In more advanced contexts, the least upper bound (supremum) and greatest lower bound (infimum) of a bounded sequence provide tight bounds: a sequence is bounded above if and only if \sup\{a_n : n \in \mathbb{N}\} < \infty, and similarly for the infimum below.^[36] For monotone sequences, boundedness is sufficient for convergence: a monotone increasing sequence that is bounded above converges to its supremum.^[37]

Advanced Sequence Structures

Subsequences

In mathematics, a subsequence of an infinite sequence (a_n)_{n=1}^\infty is obtained by selecting terms from the original sequence while preserving their relative order, specifically through a strictly increasing sequence of indices n_k such that $1 \leq n_1 < n_2 < n_3 < \cdots and the subsequence is given by (a_{n_k})_{k=1}^\infty.^[38] This construction ensures that the subsequence is itself an infinite sequence derived from a subset of the indices of the parent sequence.^[39] For example, consider the sequence of rational numbers a_n = (-1)^n / n, which alternates between positive and negative values approaching zero. The terms with even indices form the subsequence a_{2k} = 1/(2k), which is positive and decreasing to zero, while the odd-indexed terms yield a_{2k-1} = -1/(2k-1), approaching zero from the negative side.^[40] Subsequences can skip finitely or infinitely many terms, but the indices must remain strictly increasing to maintain the order.^[41] Subsequences inherit key properties from their parent sequences. If (a_n) converges to a limit L, then every subsequence (a_{n_k}) also converges to L.^[40] Similarly, if (a_n) is monotone (either non-decreasing or non-increasing), all its subsequences are monotone in the same direction.^[40] The relation of being a subsequence is reflexive (a sequence is a subsequence of itself via the identity indices) and transitive (a subsequence of a subsequence is a subsequence of the original), but not symmetric.^[38] A fundamental result is that every sequence of real numbers contains a monotone subsequence. This follows from the fact that, in any sequence, one can construct either a non-decreasing subsequence by greedily selecting terms that are at least as large as previous ones or a non-increasing one otherwise, ensuring the process continues indefinitely.^[42] The Bolzano–Weierstrass theorem provides another essential property: every bounded sequence of real numbers has at least one convergent subsequence. This theorem underpins much of real analysis, as it guarantees the existence of limit points for bounded sets derived from sequences, and its proof often relies on the monotone subsequence theorem combined with the completeness of the reals. For instance, in the bounded sequence a_n = \sin n, which does not converge, subsequences exist that converge to any value in [-1, 1] due to the density of the indices modulo $2\pi.^[42]

Constant and Arithmetic Sequences

A constant sequence is a sequence in which every term is identical, denoted as a_n = c for all n \in \mathbb{N}, where c is a fixed real number. This simplest form of sequence exhibits no variation across its terms, making it trivially convergent to c and bounded by |c|. Constant sequences serve as foundational examples in the study of sequence properties, illustrating uniform behavior without progression or oscillation. In contrast, an arithmetic sequence, also known as an arithmetic progression, is defined by a first term a_1 = a and a common difference d, such that the nth term is given by the explicit formula a_n = a + (n-1)d. This structure arises from successive additions of the fixed increment d, resulting in a linear progression. Arithmetic sequences are monotonic—increasing if d > 0, decreasing if d < 0, or constant if d = 0—and their partial sums form an arithmetic series with the formula s_n = \frac{n}{2} [2a + (n-1)d]. They appear ubiquitously in applications like modeling evenly spaced data points or calculating cumulative totals in finance. To distinguish, while constant sequences represent stasis (d = 0), arithmetic sequences generalize this to include directed change, enabling the representation of straight-line trends in discrete settings. For instance, the sequence of even numbers $2, 4, 6, \dots is arithmetic with a = 2 and d = 2, whereas the constant sequence $5, 5, 5, \dots has no such progression. Both types are finite or infinite, but their boundedness depends on d: constant sequences are always bounded, while unbounded arithmetic sequences diverge linearly. These sequences underpin more complex structures, such as in the derivation of summation formulas for higher-order progressions.

Geometric and Other Special Sequences

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the preceding term by a fixed, non-zero constant known as the common ratio r.^[43] The general term of such a sequence, starting with initial term a_1 (or a), is given by a_n = a_1 r^{n-1} for n \geq 1.^[44] This form arises directly from the recursive definition a_{n+1} = r a_n, ensuring constant ratio r = a_{n+1}/a_n for all n.^[45] The sum of the first n terms of a finite geometric sequence, denoted S_n, is S_n = a_1 \frac{1 - r^n}{1 - r} when r \neq 1.^[21] This formula derives from multiplying the sequence by r and subtracting: S_n - r S_n = a_1 (1 - r^n), then solving for S_n.^[46] For the infinite case, if |r| < 1, the series converges to S = \frac{a_1}{1 - r}, as the partial sums approach this limit due to r^n \to 0.^[47] Among other special sequences, the Fibonacci sequence stands out for its recursive structure and widespread applications. Defined by F_1 = 1, F_2 = 1, and F_n = F_{n-1} + F_{n-2} for n > 2, it generates terms 1, 1, 2, 3, 5, 8, 13, and so on.^[48] This linear homogeneous recurrence with constant coefficients leads to Binet's closed-form expression F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}, where \phi = \frac{1 + \sqrt{5}}{2} is the golden ratio, approximately 1.618.^[49] The sequence grows exponentially at rate \phi, with F_n \sim \frac{\phi^n}{\sqrt{5}} asymptotically.^[50] The harmonic sequence consists of terms whose reciprocals form an arithmetic sequence, typically exemplified by h_n = \frac{1}{n} for n \geq 1, yielding 1, 1/2, 1/3, 1/4, ....^[51] More generally, if the reciprocals $1/h_n follow an arithmetic progression with first term a and common difference d \neq 0, then h_n = \frac{1}{a + (n-1)d}.^[52] This sequence is foundational in analysis, as its partial sums H_n = \sum_{k=1}^n \frac{1}{k} (the harmonic numbers) diverge logarithmically, approximating \ln n + \gamma where \gamma \approx 0.577 is the Euler-Mascheroni constant.^[52] Other notable special sequences include the factorial sequence n!, defined recursively as $0! = 1 and n! = n \cdot (n-1)! for n \geq 1, which grows faster than exponential and underpins combinatorics and Stirling's approximation n! \approx \sqrt{2\pi n} (n/e)^n.^[53] These sequences, like geometric and Fibonacci, illustrate diverse growth patterns central to mathematical modeling.

Convergence and Limits

Limit Definition

In real analysis, the limit of a sequence captures the value that its terms approach as the index tends to infinity, providing a precise way to describe long-term behavior without requiring the sequence to attain that value at any finite step. This concept formalizes intuitive notions of "getting arbitrarily close" and underpins the study of convergence in metric spaces, particularly the real numbers.^[2] The modern rigorous definition evolved during the 19th century amid efforts to eliminate ambiguities in early calculus. Jean le Rond d'Alembert proposed an early verbal description in 1754, defining a limit as a quantity a sequence approaches without ever equaling or surpassing it, illustrated by the approximation of a circle's circumference by inscribed polygons.^[54] Augustin-Louis Cauchy advanced this in his 1821 Cours d'analyse, introducing inequalities with small positive quantities (precursors to ε) to ensure a variable differs from its limit by less than any assignable amount, laying the groundwork for limits in sequences and series.^[55] Karl Weierstrass formalized the ε-N version for sequences in his lectures during the mid-to-late 1800s, using explicit quantifiers to make proofs algebraic and universal.^[54] The standard ε-N definition states that a sequence \{a_n\}_{n=1}^\infty of real numbers converges to a limit L \in \mathbb{R} if, for every \varepsilon > 0, there exists a positive integer N such that for all n \geq N,

|a_n - L| < \varepsilon.

In logical notation, this is

\forall \varepsilon > 0, \ \exists N \in \mathbb{N}, \ \forall n \geq N, \ |a_n - L| < \varepsilon.

This ensures that the terms eventually lie within any prescribed neighborhood of L, no matter how small, with N depending on \varepsilon.^[56]^[2]^[13] Intuitively, the definition quantifies "eventual closeness": beyond some index N, the sequence remains trapped inside the interval (L - \varepsilon, L + \varepsilon), and shrinking \varepsilon forces N to grow accordingly to capture the tail's adherence. If no such L exists satisfying the condition for every \varepsilon > 0, the sequence diverges. To verify a limit, one typically assumes \varepsilon > 0, manipulates |a_n - L| algebraically to bound it by an expression involving n, and solves for N (e.g., N > f(\varepsilon) for some function f).^[56]^[2] For example, consider the sequence a_n = \frac{1}{n}. It converges to L = 0: given \varepsilon > 0, choose N = \lceil 1/\varepsilon \rceil. Then for n \geq N,

\left| \frac{1}{n} - 0 \right| = \frac{1}{n} \leq \frac{1}{N} < \varepsilon,

satisfying the definition. This illustrates how the terms diminish toward zero, with larger N needed for smaller \varepsilon.^[56]^[2]^[13]

Properties of Convergent Sequences

A convergent sequence in the real numbers possesses several fundamental properties that underpin its behavior and utility in analysis. One essential property is boundedness: every convergent sequence is bounded. Specifically, if a sequence \{a_n\} converges to a limit L, then there exists some M > 0 such that |a_n| \leq M for all n \in \mathbb{N}. This follows from the definition of convergence, where for \epsilon = 1, there is an N such that |a_n - L| < 1 for n > N, implying |a_n| < |L| + 1 for those terms, while the finite initial segment is bounded by its maximum.^[57]^[58]^[59] The limit of a convergent sequence is unique. If \{a_n\} converges to both L and L', then |L - L'| \leq |L - a_n| + |a_n - L'| < \epsilon for sufficiently large n, for any \epsilon > 0, forcing L = L'. This uniqueness ensures that the notion of "the limit" is well-defined.^[57]^[58] Convergent sequences are closed under algebraic operations, preserving convergence. For scalar multiplication, if \{a_n\} \to L and c \in \mathbb{R}, then \{c a_n\} \to c L. This is proven by bounding |c a_n - c L| = |c| \cdot |a_n - L| < |c| \epsilon. Similarly, if \{a_n\} \to L and \{b_n\} \to M, then \{a_n + b_n\} \to L + M via the triangle inequality: | (a_n + b_n) - (L + M) | \leq |a_n - L| + |b_n - M| < \epsilon. For products, \{a_n b_n\} \to L M, using the boundedness of one sequence to control the difference. Additionally, if \{a_n\} \to L \neq 0 with a_n \neq 0 for all n, then \{1/a_n\} \to 1/L, as |1/a_n - 1/L| can be bounded using a lower bound on |a_n| near L. These properties extend to differences and quotients under similar conditions.^[57]^[58]^[59] Subsequences of a convergent sequence also converge to the same limit. If \{a_n\} \to L and \{a_{n_k}\} is a subsequence, then for any \epsilon > 0, the tail condition for the original sequence applies to the subsequence indices n_k > N. In fact, the converse also holds: in metric spaces, a sequence converges to L if and only if every one of its subsequences converges to L.^[59]^[58] Order-preserving properties hold as well. If \{a_n\} \to L and a_n \geq 0 for sufficiently large n, then L \geq 0. More generally, if \{a_n\} \to L, \{b_n\} \to M, and a_n \leq b_n for large n, then L \leq M. For positive limits, compositions like square roots preserve convergence: if \{a_n\} \to L \geq 0 and a_n \geq 0 for large n, then \{\sqrt{a_n}\} \to \sqrt{L}. These follow from the squeeze theorem or direct \epsilon-bounds.^[57]^[58] The sandwich theorem provides a convergence criterion: if \{a_n\} \leq \{c_n\} \leq \{b_n\} for all n, and \{a_n\} \to L, \{b_n\} \to L, then \{c_n\} \to L. This is useful for establishing limits via inequalities, as in the convergence of \{1/n\} \to 0 by bounding between 0 and 1.^[57]

Cauchy Sequences

A Cauchy sequence is a sequence (a_n) in a metric space (X, d) such that for every \epsilon > 0, there exists a positive integer N with the property that d(a_m, a_n) < \epsilon for all integers m, n > N.^[60] This condition ensures that the terms of the sequence become arbitrarily close to each other as n increases, without necessarily specifying a particular limit point.^[61] In the context of the real numbers \mathbb{R} equipped with the standard metric d(x, y) = |x - y|, the definition simplifies to |a_m - a_n| < \epsilon for all m, n > N.^[62] The concept of Cauchy sequences provides a way to characterize convergence without reference to a limit, which is particularly useful in spaces where limits may not be immediately apparent or in abstract settings. Every convergent sequence in a metric space is Cauchy: if (a_n) converges to some L \in X, then for any \epsilon > 0, there exists N such that d(a_n, L) < \epsilon/2 for all n > N, and by the triangle inequality, d(a_m, a_n) \leq d(a_m, L) + d(a_n, L) < \epsilon for all m, n > N.^[63] However, the converse does not hold in incomplete metric spaces, such as the rational numbers \mathbb{Q} with the subspace metric from \mathbb{R}; for instance, the sequence defined by Newton's method: a_1 = 1, a_{n+1} = (a_n + 2 / a_n ) / 2 consists entirely of rationals, approximates \sqrt{2}, is Cauchy, but fails to converge in \mathbb{Q}.^[64]^[65] In the real numbers, which are a complete metric space, every Cauchy sequence converges to a limit in \mathbb{R}. This completeness property, often called the Cauchy completeness of \mathbb{R}, is a fundamental axiom equivalent to the least upper bound property and underpins the construction of \mathbb{R} from \mathbb{Q} via equivalence classes of Cauchy sequences of rationals.^[66] Specifically, two Cauchy sequences (a_n) and (b_n) in \mathbb{Q} are equivalent if \lim_{n \to \infty} (a_n - b_n) = 0, and the reals are the set of such equivalence classes with operations defined componentwise.^[64] A metric space is defined to be complete if every Cauchy sequence in it converges, making Cauchy sequences a cornerstone for studying convergence in analysis.^[61] Cauchy sequences also interact with subsequences: if (a_n) is Cauchy and (a_{n_k}) is a subsequence that converges to L, then the entire sequence (a_n) converges to L. To see this, given \epsilon > 0, choose N_1 so that d(a_m, a_n) < \epsilon/2 for m, n > N_1, and K so that d(a_{n_k}, L) < \epsilon/2 for k > K; let N = \max(N_1, n_K), then for n > N, d(a_n, L) \leq d(a_n, a_{n_K}) + d(a_{n_K}, L) < \epsilon.^[67] This property highlights their role in Bolzano-Weierstrass-type theorems, where bounded Cauchy sequences in \mathbb{R} must converge. Examples include the sequence a_n = \sum_{k=1}^n 1/k^2, which is Cauchy (and converges to \pi^2/6) due to the rapid decay of the terms, contrasting with the non-Cauchy partial sums of the harmonic series.^[68]

Non-Convergent Behaviors

In real analysis, a sequence is non-convergent, or divergent, if it does not approach a finite limit as the index tends to infinity.^[69] This divergence manifests in distinct behaviors, including unbounded growth to positive or negative infinity and oscillation without settling to a single value.^[70] One primary non-convergent behavior is divergence to infinity, where the terms grow without bound. For instance, the sequence a_n = n satisfies, for any M > 0, there exists N such that a_n > M for all n > N, indicating \lim_{n \to \infty} a_n = +\infty.^[69] Similarly, a_n = -n diverges to -\infty. Such sequences are unbounded and fail the Cauchy criterion, as the distance between terms does not shrink to zero.^[24] Another behavior is oscillation, where the sequence fluctuates indefinitely. Bounded oscillating sequences, like a_n = (-1)^n, alternate between -1 and 1, preventing convergence to any limit since subsequences converge to different points (e.g., even terms to 1, odd to -1).^[70] The Bolzano-Weierstrass theorem implies that every bounded sequence has a convergent subsequence; thus, non-convergent bounded sequences must have at least two distinct limit points, leading to oscillation.^[69] Unbounded oscillation can occur in sequences like a_n = n (-1)^n, which grows in magnitude while alternating signs.^[24] Irregular oscillation appears in sequences dense in an interval, such as a_n = \sin n, whose terms are dense in [-1, 1] due to the irrationality of \pi, ensuring no single limit exists.^[69] These behaviors highlight that non-convergence arises either from unboundedness or from multiple accumulation points, contrasting with the uniform approach to a limit in convergent sequences.^[70]

Series from Sequences

Partial Sums and Series Definition

In the context of real or complex analysis, given an infinite sequence \{a_n\}_{n=1}^\infty, the partial sums are the finite sums s_k = \sum_{n=1}^k a_n for each positive integer k, forming a new sequence \{s_k\}_{k=1}^\infty.^[71] This sequence of partial sums encapsulates the cumulative addition of the original terms up to the k-th point.^[72] An infinite series \sum_{n=1}^\infty a_n is formally defined as the sequence of these partial sums \{s_k\}_{k=1}^\infty, rather than as a direct infinite summation.^[73] The series is said to converge to a sum s if the limit \lim_{k \to \infty} s_k = s exists and is finite; otherwise, it diverges.^[74] For example, the harmonic series \sum_{n=1}^\infty \frac{1}{n} has partial sums s_k = \sum_{n=1}^k \frac{1}{n} that grow logarithmically without bound, illustrating divergence.^[71] This framework distinguishes series from mere sequences by emphasizing the behavior of their accumulating sums, which is central to convergence analysis in real analysis.^[72] Partial sums provide a concrete mechanism to study infinite processes through finite approximations, enabling the application of sequence convergence criteria to series.^[73]

Convergence Tests for Series

Convergence tests for series provide methods to determine whether an infinite series \sum_{n=1}^\infty a_n converges, meaning the sequence of partial sums s_N = \sum_{n=1}^N a_n approaches a finite limit as N \to \infty, or diverges otherwise.^[75] These tests are essential in real analysis because direct computation of partial sums is often impractical for non-trivial series. They typically apply to series with positive terms for absolute convergence or consider alternating signs for conditional convergence. Selection of a test depends on the form of a_n, such as polynomial, exponential, or factorial components.^[76] The nth-term divergence test (or divergence test) states that if \lim_{n \to \infty} a_n \neq 0 or the limit does not exist, then \sum a_n diverges. Conversely, if \lim_{n \to \infty} a_n = 0, the test is inconclusive, as the series may converge or diverge (e.g., the harmonic series \sum \frac{1}{n} diverges despite terms approaching zero). This test serves as a preliminary check before applying more advanced methods.^[77] For example, \sum_{n=1}^\infty \frac{n}{n+1} diverges because \lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0.^[78] For geometric series of the form \sum_{n=0}^\infty ar^n (with a \neq 0), the series converges to \frac{a}{1-r} if |r| < 1 and diverges if |r| \geq 1. This is derived from the partial sum formula s_N = a \frac{1 - r^{N+1}}{1 - r}, whose limit exists only when |r| < 1. Geometric series form the basis for many comparison tests. For instance, \sum_{n=1}^\infty \left(\frac{1}{2}\right)^n converges to 1.^[79]^[80] The p-series test applies to \sum_{n=1}^\infty \frac{1}{n^p}, which converges if p > 1 and diverges if p \leq 1. This result follows from the integral test and is a benchmark for comparison with similar series. The case p=1 is the divergent harmonic series, while p=2 (Basel problem) converges to \frac{\pi^2}{6}, though the test itself does not compute the sum. For example, \sum_{n=1}^\infty \frac{1}{n^{1.5}} converges since $1.5 > 1.^[81]^[75] The integral test states that if f(x) is positive, continuous, and decreasing for x \geq k with f(n) = a_n > 0 for integers n \geq k, then \sum_{n=k}^\infty a_n converges if and only if \int_k^\infty f(x) \, dx < \infty. This equates discrete summation to continuous integration. It proves the p-series result, as \int_1^\infty x^{-p} \, dx converges for p > 1. An example is \sum_{n=1}^\infty \frac{1}{n^2 + 1}, which converges because \int_1^\infty \frac{1}{x^2 + 1} \, dx = \frac{\pi}{4} < \infty.^[81]^[75] The direct comparison test for positive terms requires a known series \sum b_n: if $0 \leq a_n \leq b_n for all n sufficiently large and \sum b_n converges, then \sum a_n converges; if a_n \geq b_n \geq 0 and \sum b_n diverges, then \sum a_n diverges. Often, b_n is a geometric or p-series. For example, consider \sum_{n=1}^\infty \frac{1}{n^2 + 1}. Since $0 < \frac{1}{n^2 + 1} < \frac{1}{n^2} for all n \geq 1 and \sum_{n=1}^\infty \frac{1}{n^2} converges (p-series with p=2 > 1), \sum_{n=1}^\infty \frac{1}{n^2 + 1} converges by the direct comparison test.^[82]^[75] The limit comparison test complements the direct test: for a_n > 0, b_n > 0, let c = \lim_{n \to \infty} \frac{a_n}{b_n}. If $0 < c < \infty, then \sum a_n and \sum b_n either both converge or both diverge. This is useful when direct inequality fails but ratios approach a positive constant. For example, consider \sum_{n=1}^\infty \frac{1}{3^n + n}. Let b_n = \frac{1}{3^n}. Then c = \lim_{n \to \infty} \frac{1/(3^n + n)}{1/3^n} = \lim_{n \to \infty} \frac{3^n}{3^n + n} = 1 > 0, and since \sum b_n converges (geometric series with ratio $1/3 < 1), \sum a_n converges by the limit comparison test.^[82] The ratio test computes L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|: the series converges absolutely (hence converges) if L < 1, diverges if L > 1, and is inconclusive if L = 1. It excels for series with factorials or exponentials. For \sum_{n=1}^\infty \frac{1}{n!}, L = 0 < 1, so it converges absolutely. However, for the alternating harmonic series \sum (-1)^{n+1} \frac{1}{n}, L=1, requiring another test.^[83]^[75] The root test evaluates L = \lim_{n \to \infty} \sqrt{|a_n|}: converges absolutely if L < 1, diverges if L > 1, inconclusive if L = 1. It is effective for terms with powers, like \sum_{n=1}^\infty \left( \frac{3}{4} \right)^n, where L = \frac{3}{4} < 1, so convergent. The root test and ratio test often yield the same L via properties of limits, but the root test may be simpler for certain exponents.^[84]^[75] For alternating series \sum (-1)^{n+1} b_n with b_n > 0, the alternating series test (Leibniz test) states that if \{b_n\} is decreasing and \lim_{n \to \infty} b_n = 0, then the series converges (conditionally if \sum b_n diverges). The convergence is to a value within b_{N+1} of the partial sum s_N. For the alternating harmonic series \sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}, b_n = \frac{1}{n} decreases to 0, so it converges to \ln 2. This test does not guarantee absolute convergence.^[85] Absolute convergence occurs if \sum |a_n| converges, implying the original series converges; conditional convergence holds if it converges but not absolutely. Tests like ratio and root establish absolute convergence, while alternating series may yield conditional cases. These tests collectively cover most standard series in analysis.^[77]^[75]

Common Series Examples

One of the most fundamental examples of a series is the geometric series, defined as \sum_{k=0}^{\infty} ar^k, where a is the first term and r is the common ratio. This series converges to \frac{a}{1-r} if |r| < 1 and diverges otherwise.^[86] The convergence property arises from the partial sum formula s_n = a \frac{1 - r^{n+1}}{1 - r}, which approaches the limit as n \to \infty under the specified condition.^[79] The harmonic series, \sum_{k=1}^{\infty} \frac{1}{k}, serves as a classic example of a divergent series despite the terms approaching zero. Its divergence can be demonstrated by grouping terms, showing that partial sums exceed arbitrarily large values, such as s_{2^n} > 1 + \frac{n}{2}.^[87] In contrast, the alternating harmonic series, \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}, converges conditionally to \ln 2 by the alternating series test, highlighting the impact of sign alternation on convergence.^[88] p-series, of the form \sum_{k=1}^{\infty} \frac{1}{k^p}, provide a family of examples whose convergence depends on the parameter p. These series converge for p > 1 and diverge for p \leq 1, with the case p=1 reducing to the harmonic series; this behavior is established via the integral test, comparing the series to \int_1^{\infty} x^{-p} \, dx.^[89] For instance, the Basel problem corresponds to the convergent p-series with p=2, summing to \frac{\pi^2}{6}.^[89] Taylor series represent functions as infinite power series, exemplified by the expansion of e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}, which converges for all real x. Similarly, the sine function has the Taylor series \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, converging everywhere and providing analytic approximations near the expansion point.^[90] These series are pivotal in analysis, enabling the representation of smooth functions as limits of polynomials.^[90]

Applications in Mathematics

In Real Analysis

In real analysis, sequences form the bedrock for rigorously defining limits, continuity, and the structure of the real numbers. A sequence \{x_n\} of real numbers converges to a limit L \in \mathbb{R} if, for every \epsilon > 0, there exists N \in \mathbb{N} such that |x_n - L| < \epsilon for all n > N. This \epsilon-N definition, introduced by Karl Weierstrass in the 19th century, allows precise quantification of how terms approach the limit and is equivalent to the Cauchy criterion in complete spaces: a sequence is Cauchy if, for every \epsilon > 0, there exists N such that |x_m - x_n| < \epsilon for all m, n > N. The completeness of \mathbb{R} ensures every Cauchy sequence converges, a property absent in \mathbb{Q} (e.g., the sequence approximating \sqrt{2} diverges in \mathbb{Q} but converges in \mathbb{R}). This completeness axiom, equivalent to the least upper bound property, enables the construction of \mathbb{R} via Cauchy sequences of rationals and underpins theorems on suprema and infima.^[69]^[91] Key theorems highlight sequences' role in compactness and boundedness. The Bolzano-Weierstrass theorem states that every bounded sequence in \mathbb{R}^k has a convergent subsequence, implying every bounded infinite subset has a limit point. This result proves the Heine-Borel theorem—that closed and bounded subsets of \mathbb{R}^k are compact—and supports the extreme value theorem, ensuring continuous functions on compact sets attain maxima and minima. For monotone sequences, the monotone convergence theorem guarantees that every bounded monotone sequence converges to its supremum (if increasing) or infimum (if decreasing). These theorems are instrumental in real analysis for establishing existence results, such as the intermediate value theorem via bisection sequences, and for analyzing the topology of \mathbb{R}.^[91]^[69] Sequences of functions extend these concepts to functional analysis, distinguishing pointwise from uniform convergence. Pointwise convergence requires f_n(x) \to f(x) for each x in the domain, but it does not preserve continuity (e.g., f_n(x) = x^n on [0,1] converges pointwise to a discontinuous function). Uniform convergence, defined by \sup_{x} |f_n(x) - f(x)| \to 0, ensures the limit inherits properties like continuity: if each f_n is continuous and \{f_n\} converges uniformly to f on a set E, then f is continuous on E. Moreover, if \{f_n\} are differentiable with f_n' \to g uniformly and f_n(x_0) \to f(x_0), then f' = g. These results, including the uniform Cauchy criterion for series of functions, are crucial for justifying term-by-term differentiation or integration of power series and Fourier series, preserving integrability and differentiability in the limit. Uniform convergence on compact sets also ties back to compactness via the Arzelà-Ascoli theorem, which characterizes relatively compact families of functions.^[91]^[69] In broader applications, sequences facilitate the Riemann integral's definition through Darboux sums converging to the integral value and enable the study of metric spaces' completeness, where sequential compactness equates to every sequence having a convergent subsequence. These tools are foundational for advanced topics like measure theory and functional analysis, where sequences model approximations in Banach spaces.^[69]

In Topology

In a topological space (X, \tau), a sequence (x_n)_{n \in \mathbb{N}} in X is said to converge to a point x \in X if, for every open neighborhood U of x, there exists a positive integer N such that x_n \in U for all n > N.^[92] This generalizes the notion of convergence from metric spaces, where the topology is induced by a metric, but relies solely on the open sets without reference to distances.^[93] Constant sequences trivially converge to their repeated value in any topological space.^[94] Sequential continuity provides a way to characterize continuous functions using sequences: a function f: X \to Y between topological spaces is continuous at x \in X if, whenever a sequence (x_n) in X converges to x, the image sequence (f(x_n)) converges to f(x) in Y.^[95] This equivalence holds in general topological spaces, mirroring the situation in metric spaces, and extends to uniform continuity in appropriate settings.^[96] However, while sequences suffice to detect continuity in first-countable spaces—those where every point has a countable local basis—their limitations become apparent in broader topologies, where non-sequential points (limits not approachable by sequences) may exist.^[93] In such cases, more general tools like nets or filters are required to fully describe convergence.^[97] In first-countable spaces, sequences play a central role in determining the topology: a set is closed if and only if it contains all limits of its convergent sequences, and the closure of a set consists precisely of such limits.^[98] This property ensures that sequential convergence captures the full structure of limits, making these spaces particularly amenable to sequence-based analysis.^[94] For instance, metric spaces are first-countable, so the classical results from real analysis carry over directly.^[99] Sequential spaces, a weaker condition where the topology is determined by sequential convergence (without requiring a countable basis), further generalize this utility but may not coincide with first countability.^[99] Sequential compactness is another key concept: a topological space X is sequentially compact if every sequence in X has a subsequence that converges to a point in X.^[100] In metric spaces, this is equivalent to compactness (every open cover has a finite subcover), but the notions diverge in general topologies; for example, the ordinal space [0, \omega_1) is sequentially compact but not compact.^[101] Sequential compactness implies countable compactness and is preserved under continuous images, aiding in the study of infinite-dimensional spaces and function spaces.^[102]

In Algebra and Linear Algebra

In abstract algebra, particularly within homological algebra, sequences play a central role through the concept of exact sequences, which provide a framework for studying relationships between modules, groups, and other algebraic structures. An exact sequence is a sequence of homomorphisms f_i: A_i \to A_{i+1} between abelian groups (or more generally, modules over a ring) such that the image of each map equals the kernel of the next: \operatorname{im}(f_i) = \ker(f_{i+1}) for all i. This condition captures precise "balancing" between subspaces, allowing exact sequences to encode information about extensions, quotients, and derived functors like Tor and Ext. Seminal work in this area, including the development of long exact sequences in cohomology, stems from foundational texts in homological algebra.^[103] A prominent example is the short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, where the map f is injective (since \ker(f) = 0) and g is surjective (since \operatorname{im}(g) = C), with \operatorname{im}(f) = \ker(g). Such sequences are ubiquitous in group theory and ring theory; for instance, in the study of ideals, the sequence $0 \to I \to R \to R/I \to 0 is exact for an ideal I in a ring R, illustrating how substructures fit precisely into quotients. These sequences enable diagram chasing techniques to derive properties of the involved objects without direct computation, a method formalized in the 1940s and 1950s by pioneers like Saunders Mac Lane.^[103] In linear algebra, exact sequences specialize to the category of vector spaces over a field, where the structures simplify due to the absence of torsion and the existence of bases. For a linear transformation T: V \to W, the sequence $0 \to \ker(T) \to V \xrightarrow{T} \operatorname{im}(T) \to 0 is always short exact, reflecting the fundamental theorem of linear maps that decomposes V into kernel and image components. Crucially, every short exact sequence of vector spaces splits, meaning there exists a retraction or section such that B \cong A \oplus C in $0 \to A \to B \to C \to 0; this follows from the fact that vector spaces are free modules and admit complements via basis extensions. This splitting property underpins applications in solving systems of linear equations and understanding representations of algebras, where sequences help classify indecomposable modules.^[104]

In Combinatorics and Set Theory

In combinatorics, sequences often arise in the study of permutations and arrangements, where they model ordered selections with specific constraints. A prominent example is the longest increasing subsequence (LIS) problem, which determines the maximum length of a strictly increasing contiguous or non-contiguous subsequence in a given sequence of distinct numbers, typically drawn from a permutation of the natural numbers. This concept connects deeply to the Robinson–Schensted correspondence, a bijection between permutations and pairs of standard Young tableaux, where the length of the LIS equals the length of the longest row in the corresponding insertion tableau.^[105] The LIS has applications in analyzing the structure of random permutations, with the expected length for a random permutation of n elements asymptotically approaching $2\sqrt{n} as n grows large, highlighting the typical "disorder" in combinatorial objects.^[106] Another key structure is the Davenport–Schinzel sequence, defined over a finite alphabet of size n as a sequence without adjacent repeats and without any alternating subsequence of length exceeding a fixed parameter s, such as abab for s=2. These sequences bound the complexity of certain geometric arrangements, like the lower envelope of line segments in the plane, where the maximum length of an (n,s)-Davenport–Schinzel sequence is \Theta(n \alpha(n)) for s=3, with \alpha denoting the inverse Ackermann function, which grows extremely slowly.^[107] Introduced in the context of number theory but pivotal in extremal combinatorics, they quantify forbidden patterns and have influenced results in order types and motion planning.^[108] In set theory, sequences generalize to functions from an ordinal index set to a given universe, allowing transfinite extensions beyond countable indices. An ordinal-indexed sequence \langle x_\alpha \mid \alpha < \beta \rangle is a function f: \beta \to X for some set X, where \beta is an ordinal, enabling the construction of well-ordered chains and transfinite inductions.^[109] Such sequences are essential for defining cofinality, the smallest ordinal \mathrm{cf}(\gamma) such that there exists a cofinal map from \mathrm{cf}(\gamma) onto \gamma, meaning an increasing sequence \langle \alpha_\xi \mid \xi < \mathrm{cf}(\gamma) \rangle with supremum \gamma.^[110] For limit ordinals, regular ordinals have cofinality equal to themselves, while singular ordinals like \aleph_\omega have cofinality \omega, illustrating how sequences capture the "climbing" behavior in the ordinal hierarchy.^[111] Fundamental sequences further exemplify this in descriptive set theory and large cardinal analysis, providing a countable increasing cofinal approximation \langle \alpha_n \mid n < \omega \rangle for each ordinal \alpha < \varepsilon_0, facilitating recursive definitions and proofs of well-foundedness in impredicative constructions.^[109] These tools underpin theorems like the pressing-down lemma, which asserts that for a regressive function on a stationary set of regular cardinals, the preimage of each value is stationary, relying on diagonal intersections of sequences to preserve stationarity.^[111]

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[34]
[PDF] Monotone Sequences and Cauchy Sequences
THEOREM 17.2 (Monotone Convergence Theorem). A monotone sequence is convergent if and only if it is bounded. Proof. We have already proved (Theorem 16.13 in ...
[35]
https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/08%3A_Sequences_and_Series/8.01%3A_Sequences
[36]
[PDF] Subsequences - UCLA Department of Mathematics
A subsequence (bn) of (an) exists if there's a strictly increasing function f: N to N where bn = af(n) for all n. For example, (a 2n) is a subsequence of (an).
[37]
[PDF] 1. SeqUenCes And SUBseqUenCes
Definition 1.1. A sequence is a function a : N → R from the natural numbers to the real numbers. By setting an = a(n) ...
[38]
1. Subsequences - Advanced Analysis
Jan 17, 2024 · Given a sequence { a n } n = 1 ∞ , a subsequence is any other infinite sequence which runs through any subcollection of terms of the ...Missing: mathematical | Show results with:mathematical
[39]
Math 521 Sequences in Metric spaces
A subsequence of a given sequence xn is any other sequence yk that is of the form yk=xnk where nk is an increasing sequence of natural numbers, i.e. n1<n2<n3<⋯.Missing: mathematical | Show results with:mathematical
[40]
[PDF] Subsequences and the Bolzano-Weierstrass Theorem 1.
Sep 21, 2021 · Monotone Subsequence Theorem: Every real sequence contains a monotone subsequence. Proof: Given {xn} say xm is a “peak” if xn ≤ xm for all n>m.
[41]
[PDF] Section 1.1: Musical Notation and a Geometric Property
Jan 25, 2018 · Definition 1.1 A geometric sequence is a list of numbers where each successive number is obtained by multiplying the previous number by a ...
[42]
Math 521 Sequences in Metric spaces
A sequence xn is called a geometric sequence if xn≠0 for all n∈N, and if the ratio r=xn+1xn is the same for all n. If xn is a geometric sequence, then xn=x0rn.
[43]
https://mathcs.holycross.edu/~groberts/Courses/MA110/Handouts/Sec1.1.pdf
[44]
10-04 Geometric Sequences and Series
Deriving the formula for the sum, S, of a geometric series involves some simple factoring. S = a1 + a1 · r + a1 · r2 + a1 · ...
[45]
[PDF] Deriving the Formula for the Sum of a Geometric Series - UMD MATH
x x xx x . Finally, dividing through by 1 – x, we obtain the classic formula for the sum of a geometric series: x.
[46]
[PDF] The Fibonacci Numbers - Whitman College
Introduction. The term “Fibonacci numbers” is used to describe the series of numbers gener- ated by the pattern. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
[47]
2.2 Fibonacci Numbers
Each Fibonacci number is the sum of the previous two of them. These numbers have lots of interesting properties, and we shall look at two of them.
[48]
[PDF] Lecture Notes: Discrete Mathematics for Computer Science - Texas ...
The Rate of Growth of Fibonacci Numbers. The sequence of Fibonacci numbers Fn grows at the same rate as. (. 1 +. √. 5. 2. \n . It is asymptotically equal to. 1.
[49]
[PDF] Introduction to Series and Sequences Math 121 Calculus II
This series is called a harmonic series because its terms are in a harmonic progression. ... Definition 5 (Sequence). A sequence is an infinite list of numbers. A ...
[50]
Calculus II - Series & Sequences - Pauls Online Math Notes
Jul 11, 2023 · In this chapter we introduce sequences and series. We discuss whether a sequence converges or diverges, is increasing or decreasing, ...
[51]
[PDF] 1. Section 13.3 Geometric Sequences
Geometric Series. Definition 4.1. A geometric series is an infinite sum of a geometric sequence. Notation 4.1. S = ∞.
[52]
[PDF] Investigations Into d'Alembert's Definition of Limit (Real Analysis ...
Oct 24, 2022 · The modern definition of a limit evolved over many decades. One of the earliest attempts at a precise definition is credited to ...
[53]
[PDF] THE ORIGINS OF CAUCliY'S THEORY OF THE DERIVATIVE
Cauchy's definition of limit, with the delta-epsilon understanding that accompanied it, was the basis for the theory of convergent series he gave in the ...
[54]
[PDF] Proof Workshop Week 4: Limits and Sequences
Oct 7, 2022 · Definition 3 (Limit of a Sequence). We say that L is the limit of the sequence. {an}∞ n=1 as n tends to ∞, or equivalently, lim n→∞ an ...
[55]
[PDF] Sequences - UC Davis Math
In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some.
[56]
[PDF] Convergent Sequences
Every convergent sequence is bounded. Proof. Let (sn) be a sequence that converges to s ∈ R. Applying the definition to ε = 1, we see.
[57]
[PDF] Chapter 6 Sequences and Series of Real Numbers - Mathematics
We often use sequences and series of numbers without thinking about it. A decimal representation of a number is an example of a series, the bracketing of a ...
[58]
[PDF] Cauchy sequences
A sequence {an} is said to be Cauchy (or to be a Cauchy sequence) if for every real. number > 0, there is an integer N (possibly depending on ) for which. |an ...
[59]
2.4 Cauchy sequences - Penn Math
Subsection 2.4.1 The Cauchy property. To make this precise, here's a formal definition of what it means for a sequence to "bunch up".Missing: mathematics | Show results with:mathematics
[60]
[PDF] Math 117: Monotone and Cauchy Sequences
A sequence (sn) is increasing iff. • A sequence (sn) is decreasing iff. • A sequence (sn) is monotone iff (sn) is increasing or decreasing. We know some ...
[61]
[PDF] Math 341 Lecture #11 §2.6: The Cauchy Criterion
Every convergent sequence is a Cauchy sequence. Proof. Suppose (xn) is a convergent sequence with limit x. For > 0 there is N ∈ N such that ...
[62]
[PDF] Cauchy's Construction of R - UCSD Math
The real numbers R are the equivalence classes [(an)] of Cauchy sequences of rational numbers, as per Definition 4.1. That is, each such equivalence class is a ...
[63]
[PDF] II.3. Sequences and Completeness
Oct 8, 2017 · By the Triangle Inequality, all convergent sequences are Cauchy. However, the convergence of Cauchy sequences requires the Completeness of R.
[64]
[PDF] 18.100A Fall 2020 Lecture 10: The Completeness of the Real ...
Theorem 6. If {xn} is Cauchy and a subsequence {xnk } converges, then {xn} converges. Proof: Suppose that {xnk } is a subsequence of {xn} such that limk→∞ xnk.
[65]
Math 413 – Cauchy sequences and Bolzano Weierstrass
Oct 8, 2020 · A Cauchy sequence has its terms getting very close together eventually, but we are not necessarily given that the sequence converges. In R, a ...
[66]
[PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
The set N is closed, but it is not sequentially compact since the sequence (n) in N has no convergent subsequence. (Every subsequence diverges to infinity.) ...
[67]
4.1: Sequences
### Summary of Non-Convergent Sequences, Divergent Behaviors, Oscillation, and Divergence to Infinity
[68]
Calculus II - Series - The Basics - Pauls Online Math Notes
Nov 16, 2022 · A sequence is a list of numbers written in a specific order while an infinite series is a limit of a sequence of finite series and hence, if it exists will be ...
[69]
[PDF] Series - UC Davis Math
A series is an infinite sum. It converges if its sequence of partial sums converges, and its sum is the limit of those partial sums.
[70]
[PDF] Chapter 0. Series of numbers
... define the nth partial sum sn to be the sum of the first n terms in the series. In this way we get a sequence s1,s2,s3, ··· called the sequence of partial sums.<|control11|><|separator|>
[71]
4.1. Series and Convergence - Real Analysis - MathCS.org
A series is the result of an infinite addition - which we do not yet know how to handle - each partial sum is the sum of finitely many terms only.
[72]
[PDF] RES.18-001 Calculus (f17), Chapter 10: Infinite Series
The third has plus signs and also converges to ln 2. These will be examples for a major topic in infinite series— tests for convergence. For the first time in ...Missing: reliable | Show results with:reliable
[73]
Calculus II - Strategy for Series - Pauls Online Math Notes
Nov 16, 2022 · In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or ...
[74]
Calculus II - Convergence/Divergence of Series
Aug 13, 2024 · A series is convergent if its partial sums have a finite limit, and divergent if the limit doesn't exist or is infinity. If the limit of the ...
[75]
[PDF] Testing for Convergence or Divergence of a Series - CSUSM
Some series will “obviously” not converge—recognizing these can save you a lot of time and guesswork. Test for Divergence. If. 0 lim. ≠. ∞. → n.
[76]
Calculus II - Special Series
### Summary of p-Series and Special Series Tests
[77]
Strategies for Testing Series
If you see that the terms an do not go to zero, you know the series diverges by the Divergence Test. · If a series is a p-series, with terms 1np, we know it ...
[78]
Calculus II - Integral Test - Pauls Online Math Notes
Nov 16, 2022 · Using the p p -series test makes it very easy to determine the convergence of some series. Example 3 Determine if the following series are ...
[79]
Calculus II - Comparison Test/Limit Comparison Test
### Comparison Test
[80]
Calculus II - Ratio Test
### Summary of the Ratio Test for Series Convergence
[81]
Calculus II - Root Test - Pauls Online Math Notes
Nov 16, 2022 · In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. The Root Test can ...
[82]
Calculus II - Alternating Series Test
### Summary of Alternating Series Test
[83]
Geometric Series -- from Wolfram MathWorld
A geometric series sum_(k)a_k is a series for which the ratio of each two consecutive terms a_(k+1)/a_k is a constant function of the summation index k.
[84]
Harmonic Series -- from Wolfram MathWorld
The series sum_(k=1)^infty1/k is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function 1/x.
[85]
Alternating Harmonic Series -- from Wolfram MathWorld
The alternating harmonic series is the series sum_(k=1)^infty((-1)^(k-1))/k=ln2, which is the special case eta(1) of the Dirichlet eta function eta(z).
[86]
p-Series -- from Wolfram MathWorld
A shorthand name for a series with the variable k taken to a negative exponent, e.g., sum_(k=1)^(infty)k^(-p), where p>1. p-series are given in closed form ...
[87]
Taylor Series -- from Wolfram MathWorld
A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x=a.
[88]
[PDF] Rudin (1976) Principles of Mathematical Analysis.djvu
This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first-year students who study ...
[89]
[PDF] Chapter 4: Topological Spaces - UC Davis Math
Let X and Y be a topological spaces. Definition 4.5 A sequence ( x n) in X converges to a limit x ∈ X if for every.
[90]
[PDF] Topology - I
Definition 3.1 A sequence {xn} of points of a space X converges to a point x if for any neighborhood U of x, there is an N > 0 such that whenever n>N, xn ∈ U.
[91]
[PDF] 5. Sequences, weak T-axioms, and first countability
3. Trivially, a constant sequence x, x, x, x, . . . converges to x in any topological space. The same is true of a sequence that is constant on some tail. 4.
[92]
[PDF] A Review of General Topology. Part 3: Sequential Convergence
Definition 3 (Continuous for Sequences). Let f : X → Y be a function between topological spaces. We say that f is continuous for sequences at a point x ∈ X.
[93]
[PDF] A.7 Convergence and Continuity in Topological Spaces
In metric spaces, convergence is defined with respect to sequences indexed by the natural numbers (Definition A.2). In a general topological space, conver-.
[94]
[PDF] Sequences and nets in topology - arXiv
Jun 23, 2010 · sequences in topology, their use is not suitable in the study of ... Munkres. Topology. Prentice Hall, second edition, 2000. [18] ...
[95]
[PDF] Functional Analysis, Math 7320 Lecture Notes from September 15 ...
Sep 15, 2016 · Let (X, τ) be a first countable topological space. Then. (1) for Y ⊂ X, a ∈ Y if and only if there exists (xn)n∈N in Y with xn ...<|control11|><|separator|>
[96]
[PDF] On Certain Topological Spaces In Which Sequential Convergence ...
There is a hierarchy of topological spaces specified by the convergence of sequences. Best known of these are the first countable and sequential spaces.<|control11|><|separator|>
[97]
[PDF] 17. Tychonoff's theorem, and more on compactness
Definition 4.1. A topological space (X,T ) is called sequentially compact if every sequence in X has a convergent subsequence. As the name implies, this is an ...<|separator|>
[98]
[PDF] Various Notions of Compactness
Jun 18, 2012 · Definition 5 (Sequentially Compact) K ⊆ X is sequentially compact if every sequence of it has a con- vergent subsequence. As the name suggests, ...
[99]
[PDF] LIST FOUR Topic 1: Countable compactness and sequential ...
X is sequentially compact if any sequence on X admits a convergent subse- quence.
[100]
[PDF] AN INTRODUCTION TO HOMOLOGICAL ALGEBRA
This book is an introduction to homological algebra, covering topics such as complexes, derived functors, and Tor and Ext.
[101]
Short Exact Sequence -- from Wolfram MathWorld
M is projective, and all short exact sequences. 0-->M-->B-->C-->. (3). are split iff M is injective. A short exact sequence of vector spaces is always split.
[102]
[PDF] LONGEST INCREASING AND DECREASING SUBSEQUENCES
This paper deals with finite sequences of integers. Typical of the problems we shall treat is the determination of the number of sequences of length n,.
[103]
[PDF] LONGEST INCREASING SUBSEQUENCES: FROM PATIENCE ...
Jul 21, 1999 · The other theme is a card game, patience sorting. This game provides an elementary context in which the longest increasing subsequence arises.
[104]
[PDF] Davenport{Schinzel Sequences and Their Geometric Applications
Feb 5, 1998 · Davenport{Schinzel sequences, introduced by H. Davenport and A. Schinzel in the 1960s, are interesting and powerful combinatorial structures ...
[105]
Davenport–Schinzel Sequences and Their Geometric Applications
Davenport–Schinzel sequences have become a useful and powerful tool for solving numerous problems in discrete and computational geometry, usually by showing ...
[106]
[PDF] Ordinal numbers - Cornell Mathematics
To this end we define by induction a 1–1 transfinite sequence xα ∈ X, indexed by an initial segment of the ordinals. Assume inductively that α is an ordinal ...
[107]
Section 3.7 (000E): Cofinality—The Stacks project
3.7 Cofinality. A cofinal subset $S$ of a well-ordered set $T$ is a subset $S \subset T$ such that $\forall t \in T \exists s\in S (t \leq s)$.
[108]
[PDF] cofinality spectrum theorems in model theory, set ... - UChicago Math
This shows the existence of a maximum class in. Keisler's order, which can be characterized set-theoretically as the set of complete countable first-order ...<|separator|>