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Radius of convergence

The radius of convergence of a power series \sum_{n=0}^\infty a_n (z - c)^n, where a_n are complex coefficients and c is the center, is the nonnegative real number R (possibly $0 or \infty) such that the series converges absolutely for all z in the open disk |z - c| < R and diverges for |z - c| > R.^[1] In the real-variable case, where z = x \in \mathbb{R}, this corresponds to convergence on the open interval (c - R, c + R).^[2] The value of R determines the region of convergence but does not specify behavior on the boundary circle or endpoints |z - c| = R, where convergence must be checked separately using other tests.^[3] The radius R can be computed using the root test formula \frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}, which always exists and provides the precise value even when limits of ratios do not.^[4] Alternatively, if the limit \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L exists and is finite, then \frac{1}{R} = L, derived from the ratio test applied to the series terms.^[2] For R = 0, the series converges only at the center z = c; for R = \infty, it converges everywhere in the complex plane.^[3] These formulas extend naturally to real power series \sum_{n=0}^\infty a_n (x - c)^n.^[5] Within the disk of convergence, the power series represents an analytic function, meaning the sum is holomorphic (complex differentiable) and can be differentiated or integrated term by term.^[1] The radius R is intrinsically linked to the distance from c to the nearest singularity of the represented function in the complex plane, limiting the extent of uniform convergence on compact subsets inside the disk.^[6] This property underscores the radius's role in complex analysis, where power series expansions are central to studying holomorphic functions and their domains.^[1] In applications, such as solving differential equations or approximating functions, determining R ensures the validity of the series representation over desired regions.^[6]

Core Concepts

Definition

In complex analysis, a power series centered at a point c \in \mathbb{C} is an infinite series of the form \sum_{n=0}^{\infty} a_n (z - c)^n, where a_n are complex coefficients and z is a complex variable.^[7] The radius of convergence R, where $0 \leq R \leq \infty, is defined as the largest number such that the series converges for all z satisfying |z - c| < R and diverges for all z with |z - c| > R.^[7]^[8] This radius characterizes the domain of convergence in the complex plane, forming an open disk centered at c with radius R, often called the disk of convergence.^[7] Within this disk of convergence, the power series exhibits absolute convergence, meaning \sum_{n=0}^{\infty} |a_n (z - c)^n| < \infty for every z in the interior, which ensures the series sums to a holomorphic (analytic) function on that open set.^[7]^[8] On the boundary circle |z - c| = R (when $0 < R < \infty), convergence behavior can vary and is not guaranteed, potentially converging at some points and diverging at others.^[7] Although power series are fundamentally defined over the complex numbers to capture their full analytic properties, the concept restricts naturally to the real line by considering only real z, yielding an interval of convergence (c - R, c + R).^[8] Special cases delineate the extent of applicability: if R = 0, the series converges solely at the center z = c and nowhere else, rendering it non-analytic on any open disk; conversely, if R = \infty, the series converges for all z \in \mathbb{C}, defining an entire function holomorphic everywhere in the plane.^[7]^[8] These properties underscore the radius of convergence as a fundamental invariant that delimits the region where the power series represents a well-behaved analytic function.^[7]

Power series context

A power series is an infinite series of the form \sum_{n=0}^{\infty} a_n (z - c)^n, where c \in \mathbb{C} is the center, a_n \in \mathbb{C} are the coefficients, and z \in \mathbb{C} is the variable.^[8] This representation allows functions to be expanded around a point c, with the series potentially converging to the function within a certain region. The radius of convergence R \geq 0 delineates the open disk |z - c| < R where the series converges pointwise, and diverges for |z - c| > R. Within this disk of convergence, the power series exhibits strong convergence properties, including uniform convergence on any compact subset K such that |z - c| \leq r < R for some r > 0. This uniform convergence implies that the partial sums approximate the sum function continuously and allows term-by-term differentiation and integration, preserving analyticity inside the disk. Such behavior underscores the power series' utility in representing holomorphic functions locally. Power series are intimately linked to Taylor series expansions of functions that are holomorphic (complex differentiable) in a neighborhood of the center c. For an analytic function f at c, its Taylor series \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (z - c)^n has coefficients a_n = \frac{f^{(n)}(c)}{n!}, and the radius of convergence matches the distance from c to the nearest singularity of f.^[9] This connection highlights how the radius governs the extent of local analytic continuation via power series. The radius R can be characterized explicitly via Hadamard's formula: \frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}, providing a direct means to assess convergence from the coefficients alone.^[10] This formula, also known as the Cauchy-Hadamard theorem, encapsulates the asymptotic growth of the coefficients in determining the series' domain of convergence.^[8]

Computation Techniques

Ratio test method

The ratio test provides a practical method for determining the radius of convergence R of a power series \sum_{n=0}^{\infty} a_n (z - c)^n by examining the limit of the ratios of consecutive coefficients. Specifically, if the limit \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| = L exists and is finite (including L = 0 or L = \infty), then R = L.^[11]^[4] This result derives from the ratio test for absolute convergence of series, which states that for a series \sum b_n, if \lim_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| = M < 1, then the series converges absolutely. Applying this to the power series with b_n = a_n (z - c)^n, the ratio becomes \left| \frac{b_{n+1}}{b_n} \right| = \left| \frac{a_{n+1}}{a_n} \right| |z - c|. Taking the limit yields \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| |z - c| < 1, so |z - c| < \frac{1}{\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|} = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| when the limit exists. This establishes the radius R as the distance from the center c within which the series converges absolutely./08%3A_Sequences_and_Series/8.06%3A_Power_Series)^[2] The ratio test is equivalent to the root test in cases where the coefficient limit exists, as \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \limsup_{n \to \infty} |a_n|^{1/n} under these conditions, aligning the radii computed by both methods.^[4] The method applies only when the limit \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| exists; if the limit does not exist or oscillates, the ratio test fails, and alternative approaches such as the root test must be used. In such cases, the radius can still be found via \frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}.^[11]/08%3A_Sequences_and_Series/8.06%3A_Power_Series) For a generic example, consider a power series where the coefficients satisfy \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| = 3. The computation proceeds by evaluating the limit directly from the sequence of ratios, yielding R = 3, so the series converges absolutely for |z - c| < 3.^[2]

Root test method

The root test provides a general method for determining the radius of convergence R of a power series \sum_{n=0}^{\infty} a_n (z - c)^n through the formula

R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}},

where the reciprocal of the limit superior (lim sup) yields the radius, with the conventions that R = \infty if the lim sup is 0 and R = 0 if the lim sup is \infty.^[8] This approach guarantees absolute convergence for |z - c| < R and divergence for |z - c| > R.^[8] The use of lim sup in the root test addresses irregular or oscillatory growth in the coefficients a_n, where the ordinary limit \lim_{n \to \infty} |a_n|^{1/n} may fail to exist; lim sup captures the largest accumulation point of the sequence |a_n|^{1/n}, ensuring a well-defined value even for sequences with multiple limit points or erratic behavior.^[8]^[12] For instance, if |a_n|^{1/n} oscillates but clusters around certain values, the lim sup identifies the supremum of these cluster points, providing the precise boundary for convergence.^[12] In comparison to the ratio test, the root test succeeds in cases where the limit \lim_{n \to \infty} |a_{n+1}/a_n| does not exist due to irregular coefficient ratios, as the nth-root structure often stabilizes the growth rate more robustly.^[8] While the ratio test is simpler when its limit exists, the root test's reliance on lim sup makes it more versatile for general coefficient sequences.^[8] When computing the exact lim sup analytically proves challenging, practical estimation involves generating the sequence |a_n|^{1/n} for large n, identifying upper bounds on tail suprema via \sup_{k \geq n} |a_k|^{1/k} for increasing n, and observing convergence of these values to approximate the lim sup; this numerical approach leverages the monotonicity of the tail suprema to bound R.^[8]^[12]

Illustrative Examples

Geometric series example

The geometric series \sum_{n=0}^{\infty} z^n provides a fundamental example of a power series with radius of convergence R = 1. To determine this radius, apply the ratio test: consider \lim_{n \to \infty} \left| \frac{z^{n+1}}{z^n} \right| = |z|, so the series converges when |z| < 1 and diverges when |z| > 1. Within the open unit disk |z| < 1 in the complex plane, the series converges absolutely to the function \frac{1}{1 - z}, which is holomorphic everywhere inside this disk. This convergence region forms a circular disk centered at the origin with radius 1, illustrating the typical geometry of power series convergence domains. On the boundary |z| = 1, the series diverges at every point, including z = 1, because the general term z^n does not tend to zero (as |z^n| = 1). For instance, at z = 1, the series becomes \sum_{n=0}^{\infty} 1, which clearly diverges to infinity.

Logarithmic series example

The power series \sum_{n=1}^\infty \frac{z^n}{n} converges to -\log(1 - z) within the unit disk |z| < 1.^[13] The coefficients a_n = 1/n arise from the Taylor expansion of the logarithm function centered at z = 0. This expansion is derived by starting with the geometric series \frac{1}{1 - z} = \sum_{n=0}^\infty z^n for |z| < 1, noting that the derivative of \log(1 - z) is -\frac{1}{1 - z}, and integrating term by term: \log(1 - z) = -\int_0^z \sum_{n=0}^\infty t^n \, dt = -\sum_{n=0}^\infty \int_0^z t^n \, dt = -\sum_{n=1}^\infty \frac{z^n}{n}, where the constant of integration is zero by evaluation at z = 0.^[13] The radius of convergence R is computed using the root test formula R = 1 / \limsup_{n \to \infty} |a_n|^{1/n}. Substituting a_n = 1/n gives |a_n|^{1/n} = (1/n)^{1/n} = n^{-1/n}. Since \lim_{n \to \infty} n^{1/n} = 1 (proved by writing n^{1/n} = e^{(\ln n)/n} and noting (\ln n)/n \to 0 by L'Hôpital's rule, so e^0 = 1), it follows that \lim_{n \to \infty} (1/n)^{1/n} = 1, hence R = 1.^[14]^[15] Inside the unit disk |z| < 1, the series equals -\log(1 - z); outside |z| > 1, the series diverges.^[16] (pp. 180–182) The point z = 1 is a branch point singularity of -\log(1 - z), located on the boundary of the convergence disk.^[16] (pp. 46–48)

Complex Domain Analysis

Holomorphic extension

A power series with a positive radius of convergence defines a holomorphic function within its open disk of convergence in the complex plane.^[17] Specifically, if the series converges uniformly on compact subsets of the disk, the sum is holomorphic there, and term-by-term differentiation yields the derivative, preserving holomorphy.^[17] This representation is unique, as the coefficients are determined by the function's values via Cauchy's integral formula.^[17] The radius of convergence corresponds to the distance from the center to the nearest singularity of the function in the complex plane, marking the maximal disk where the series converges to a holomorphic function.^[6] Beyond this radius, the series diverges, and the function may have a singularity preventing holomorphy at that point.^[6] For instance, the geometric series \sum_{n=0}^\infty z^n has radius 1, limited by the pole at z=1.^[6] The principle of analytic continuation allows extending the holomorphic function beyond the initial disk of convergence to a larger domain, provided no singularities obstruct the path.^[18] This extension is unique by the identity theorem: if two analytic continuations agree on a connected open set intersecting the original domain, they coincide everywhere in the larger domain.^[18] For example, the geometric series \sum_{n=0}^\infty z^n = \frac{1}{1-z} inside |z|<1 continues analytically to \mathbb{C} \setminus \{1\}, avoiding the singularity at z=1.^[18] Certain entire functions, holomorphic everywhere in the complex plane, admit power series with infinite radius of convergence, implying no singularities.^[8] The exponential function e^z = \sum_{n=0}^\infty \frac{z^n}{n!} exemplifies this, converging for all z \in \mathbb{C} with no singularities, as confirmed by the ratio test yielding R = \infty.^[8] Thus, its Taylor series around any point represents the function globally without extension needed.^[8]

Boundary behavior

For a power series \sum a_n (z - c)^n with radius of convergence R > 0, the behavior on the boundary circle |z - c| = R is highly variable: the series may converge at none, some, or all points on this circle, and convergence, when it occurs, need not be uniform. Inside the open disk |z - c| < R, convergence is absolute and uniform on compact subsets, but on the boundary, absolute convergence is exceptional and typically limited to specific cases where the coefficients decay rapidly enough for \sum |a_n| R^n < \infty. For instance, the series \sum_{n=1}^\infty \frac{(z - c)^n}{n^2} converges absolutely at every boundary point since \sum \frac{R^n}{n^2} < \infty, yielding a continuous extension to the closed disk. In contrast, conditional convergence—where the series converges but not absolutely—is more common on the boundary, as seen in \sum_{n=1}^\infty \frac{(z - c)^n}{n}, which converges at all boundary points except z = c + R (where it diverges like the harmonic series), but \sum \frac{R^n}{n} = \infty.^[19] Abel's theorem provides a key insight into the continuity of the power series function across the boundary along radial paths. Specifically, if the series converges to a value S at a boundary point \zeta with |\zeta - c| = R, then the radial limit satisfies \lim_{r \to 1^-} f(r (\zeta - c) + c) = S, where f(z) is the sum inside the disk; this holds even if the function does not extend holomorphically beyond \zeta. This theorem ensures that the boundary value, when the series converges, matches the continuous radial approach from within the disk, facilitating connections to summability methods. For example, in the conditionally convergent case \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \log 2 at the boundary point corresponding to z = c - R, Abel's theorem confirms the radial limit equals this alternating harmonic sum.^[20] The irregular convergence on the boundary often stems from singularities of the analytic function defined by the series. The radius R is precisely the distance from c to the nearest singularity in the complex plane, and such singularities on or near the circle |z - c| = R dictate points of divergence: at a singularity \zeta with |\zeta - c| = R, the series necessarily diverges, as the function cannot be analytically continued through that point. This phenomenon links power series boundary behavior to Fourier series representations, where the boundary function (when it exists) admits a trigonometric expansion, reflecting the periodic nature of the circle and highlighting conditional convergence patterns akin to those in L^2 integrable functions on the unit circle. For instance, the logarithm series \sum \frac{z^n}{n} has a branch point singularity at z = 1 (assuming c=0, R=1), causing divergence there while converging conditionally elsewhere on the boundary via Dirichlet's test.^[19]^[21]

Extensions and Variations

Rate of convergence

The rate of convergence of a power series \sum_{n=0}^\infty a_n (z - c)^n with radius of convergence R > 0 describes the speed at which the partial sums s_N(z) = \sum_{n=0}^N a_n (z - c)^n approximate the sum function f(z) inside the open disk |z - c| < R. For |z - c| < r < R, the remainder term after N terms, R_N(z) = f(z) - s_N(z) = \sum_{n=N+1}^\infty a_n (z - c)^n, satisfies the error estimate

|R_N(z)| \leq M \left( \frac{|z - c|}{r} \right)^{N+1},

where M = \sum_{n=N+1}^\infty |a_n| r^n is the tail of the absolute series at radius r. This bound arises by noting that |R_N(z)| \leq \sum_{n=N+1}^\infty |a_n| r^n \cdot \rho^n = \rho^{N+1} \sum_{k=0}^\infty |a_{N+1+k}| r^{N+1+k} \leq M \rho^{N+1}, with \rho = |z - c|/r < 1. The convergence rate is geometric with ratio \rho < 1, so the error decays exponentially as N increases. This rate depends on the distance from the center c: it is faster near c (where \rho is small for fixed r) and slower as |z - c| approaches the boundary of the disk (where \rho nears 1, requiring larger N for comparable accuracy). Asymptotic rates of convergence inside the disk are tied to the growth of the coefficients a_n, as the radius satisfies R = 1 / \limsup_{n \to \infty} |a_n|^{1/n}, determining the minimal \rho achievable near the boundary.

Dirichlet series abscissa

A Dirichlet series is a series of the form \sum_{n=1}^\infty \frac{a_n}{n^s}, where a_n are complex coefficients and s = \sigma + it is a complex variable with real part \sigma and imaginary part t. The abscissa of convergence \sigma_c is the infimum of the real numbers \sigma such that the series converges for all s with \operatorname{Re}(s) > \sigma. This defines a right half-plane of convergence in the complex plane, where the series converges absolutely for \operatorname{Re}(s) > \sigma_a with \sigma_a \geq \sigma_c, and \sigma_a being the abscissa of absolute convergence.^[22] The value of \sigma_c can be determined using Cahen's formula:

\sigma_c = \limsup_{n \to \infty} \frac{\log \left| \sum_{k=1}^n a_k \right|}{\log n},

where the partial sums S_n = \sum_{k=1}^n a_k are considered; if the series of partial sums diverges, this limit superior gives the precise boundary. This formula arises from analyzing the growth of the partial sums relative to the logarithmic scale of n, providing a direct computational tool for specific coefficients a_n.^[23] This concept of the abscissa \sigma_c serves as an analogue to the radius of convergence R for power series, where convergence occurs inside a disk |z| < R in the complex plane, but for Dirichlet series, the region is instead a half-plane \operatorname{Re}(s) > \sigma_c. In number theory, Dirichlet series play a central role, such as the Riemann zeta function \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}, for which a_n = 1, the partial sums grow like n, and thus \sigma_c = 1, marking the boundary where the series converges for \operatorname{Re}(s) > 1 and relates to the distribution of prime numbers via the Euler product.^[22]^[24]

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