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Stolz–Cesàro theorem

The Stolz–Cesàro theorem is a fundamental result in mathematical analysis that provides a method for determining the limit of a quotient of two sequences when direct evaluation is indeterminate, analogous to L'Hôpital's rule for continuous functions.^[1] Specifically, given sequences \{a_n\} and \{b_n\} of real numbers where \{b_n\} is strictly increasing and unbounded (i.e., b_n \to \infty as n \to \infty), if the limit \lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L exists (with L \in \mathbb{R} \cup \{\pm \infty\}), then \lim_{n \to \infty} \frac{a_n}{b_n} = L.^[2] This theorem, named after Otto Stolz and Ernesto Cesàro who independently developed it in the late 19th century, addresses forms like \infty/\infty or $0/0 in discrete settings and requires no assumptions on the monotonicity of \{a_n\}.^[3] The theorem's significance lies in its ability to simplify limit calculations for partial sums and averages, making it indispensable for proving convergence in sequences and series.^[1] For instance, when b_n = 1 for all n, it reduces to the additive Cesàro theorem, which equates the limit of the sequence to the limit of its arithmetic means if the latter exists.^[1] It also extends to multiplicative cases via logarithmic transformations for products of positive terms and generalizes L'Hôpital's rule to sequences by considering difference quotients.^[1] Broader variants apply when both sequences tend to zero or in more general monotone settings, enhancing its utility in advanced topics like asymptotic analysis and probability.^[3] Originally introduced by Otto Stolz in his 1879 paper "Ueber die Grenzwerthe der Quotienten" in Mathematische Annalen, the theorem was later independently proven by Ernesto Cesàro, solidifying its place in analysis.^[3] Today, it remains a cornerstone tool in calculus textbooks and research, frequently employed to evaluate limits involving harmonic series, power sums, and stochastic processes.^[1]

Theorem Statements

∞/∞ Case

The Stolz–Cesàro theorem addresses indeterminate forms of type ∞/∞ for sequences, serving as a discrete counterpart to l'Hôpital's rule for continuous functions, where the limit of a quotient is inferred from the limit of the quotient of consecutive differences when the denominator diverges to infinity.^[4]^[5] Consider two sequences of real numbers (a_n)_{n=1}^\infty and (b_n)_{n=1}^\infty, where (b_n) is strictly increasing and \lim_{n \to \infty} b_n = +\infty. Suppose that the limit

\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L

exists, where L \in \mathbb{R} \cup \{\pm \infty\}. Then,

\lim_{n \to \infty} \frac{a_n}{b_n} = L.

^[6]^[5] This formulation assumes b_{n+1} - b_n > 0 for all n to ensure the differences are positive, consistent with the strict monotonicity of (b_n).^[6] The theorem extends symmetrically to the 0/0 case by considering limits as both sequences approach zero.^[5]

0/0 Case

The Stolz–Cesàro theorem provides a criterion for evaluating the limit of the quotient of two sequences (a_n) and (b_n) that both converge to zero, analogous to L'Hôpital's rule for continuous functions in the indeterminate form $0/0.^[7] Suppose (a_n)_{n=1}^\infty and (b_n)_{n=1}^\infty are sequences of real numbers such that \lim_{n \to \infty} a_n = 0 and \lim_{n \to \infty} b_n = 0. Assume further that (b_n) is strictly decreasing for sufficiently large n and b_n > 0 for all n. If the limit

\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L

exists (where L is a real number or \pm \infty), then

\lim_{n \to \infty} \frac{a_n}{b_n} = L.

^[7] If instead (b_n) is strictly increasing and convergent to 0, the theorem can be applied by considering the negated sequences (-a_n) and (-b_n), which transform the situation into a decreasing case with the limit sign adjusted accordingly.^[7] The requirement that (b_n) be strictly monotone and convergent to zero ensures that the differences b_{n+1} - b_n have consistent signs, allowing the successive differences to telescope appropriately in the analysis, in contrast to the \infty/\infty case where (b_n) is typically strictly increasing to infinity without bound.^[7] The \infty/\infty variant is more commonly invoked for unbounded sequences.^[7]

Proofs

∞/∞ Case Proof

Consider sequences (a_n) and (b_n) of real numbers where (b_n) is strictly increasing with b_n \to \infty as n \to \infty, and assume \lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L where L \in \mathbb{R}. To prove \lim_{n \to \infty} \frac{a_n}{b_n} = L, define the auxiliary sequence c_n = a_n - L b_n. Then,

\frac{c_{n+1} - c_n}{b_{n+1} - b_n} = \frac{(a_{n+1} - a_n) - L (b_{n+1} - b_n)}{b_{n+1} - b_n} = \frac{a_{n+1} - a_n}{b_{n+1} - b_n} - L \to 0

as n \to \infty. (Stolz, 1885, p. 173) It suffices to show that if \lim_{n \to \infty} \frac{c_{n+1} - c_n}{b_{n+1} - b_n} = 0, then \lim_{n \to \infty} \frac{c_n}{b_n} = 0. Let \varepsilon > 0. There exists N \in \mathbb{N} such that for all n \geq N,

\left| \frac{c_{n+1} - c_n}{b_{n+1} - b_n} \right| < \varepsilon,

so |c_{n+1} - c_n| < \varepsilon (b_{n+1} - b_n). For m > k \geq N,

|c_m - c_k| = \left| \sum_{j=k}^{m-1} (c_{j+1} - c_j) \right| \leq \sum_{j=k}^{m-1} |c_{j+1} - c_j| < \varepsilon \sum_{j=k}^{m-1} (b_{j+1} - b_j) = \varepsilon (b_m - b_k).

Fix k \geq N. Then for m > k,

\frac{c_m}{b_m} = \frac{c_k}{b_m} + \frac{c_m - c_k}{b_m},

and

\left| \frac{c_m - c_k}{b_m} \right| < \varepsilon \frac{b_m - b_k}{b_m} < \varepsilon.

As m \to \infty, \frac{c_k}{b_m} \to 0 since b_m \to \infty and c_k is fixed. Thus,

\limsup_{m \to \infty} \left| \frac{c_m}{b_m} \right| \leq \varepsilon.

Since \varepsilon > 0 is arbitrary, \lim_{m \to \infty} \frac{c_m}{b_m} = 0, so \lim_{n \to \infty} \frac{a_n}{b_n} = L. (Stolz, 1885, pp. 174–175) The expression \frac{a_n}{b_n} can be viewed as arising from summation: a_n = a_N + \sum_{j=N}^{n-1} (a_{j+1} - a_j) and b_n = b_N + \sum_{j=N}^{n-1} (b_{j+1} - b_j) for fixed N large, so

\frac{a_n}{b_n} = \frac{a_N + \sum_{j=N}^{n-1} \frac{a_{j+1} - a_j}{b_{j+1} - b_j} (b_{j+1} - b_j)}{b_n},

a weighted average of the difference quotients \frac{a_{j+1} - a_j}{b_{j+1} - b_j} with weights b_{j+1} - b_j > 0. As n \to \infty, the initial term \frac{a_N}{b_n} \to 0 and the quotients converge to L, so the average converges to L by the assumed monotonicity and divergence of b_n. (Stolz, 1885, p. 175) For the case L = +\infty, assume \lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = +\infty. For any M > 0, there exists N such that for n \geq N, \frac{a_{n+1} - a_n}{b_{n+1} - b_n} > M. Then for m > k \geq N,

a_m - a_k > M (b_m - b_k),

\frac{a_m}{b_m} > M \left(1 - \frac{b_k}{b_m}\right) + \frac{a_k}{b_m}.

As m \to \infty, the right side exceeds M - \varepsilon for large m. Thus, \liminf_{m \to \infty} \frac{a_m}{b_m} \geq M. Since M is arbitrary, \lim_{n \to \infty} \frac{a_n}{b_n} = +\infty. The case L = -\infty follows similarly by replacing > with < and M with -M. (Stolz, 1885, p. 173) The key inequality underlying the bounds is, for large n < m and \varepsilon > 0, |a_m - a_n| < ( |L| + \varepsilon ) (b_m - b_n ) + C where C accounts for initial terms, which becomes negligible relative to b_m as m \to \infty. (Stolz, 1885, p. 174)

0/0 Case Proof

The 0/0 case of the Stolz–Cesàro theorem addresses limits of the form \lim_{n \to \infty} \frac{a_n}{b_n} where both a_n \to 0 and b_n \to 0, with b_n strictly monotone (typically decreasing and positive for the standard formulation). The proof adapts the summation technique from the \infty/\infty case by considering differences over a range from a fixed large index to infinity, leveraging the convergence to zero to establish the limit of the ratio via bounding arguments. Assume without loss of generality that b_n > 0 is strictly decreasing to 0 (the increasing case follows similarly with sign adjustments), and \lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L \in \mathbb{R}.^[8] To verify the conditions and derive the limit, fix \varepsilon > 0. By the assumption on the difference quotient, there exists N \in \mathbb{N} such that for all n > N,

\left| \frac{a_{n+1} - a_n}{b_{n+1} - b_n} - L \right| < \frac{\varepsilon}{2}.

Since b_n is strictly decreasing, b_{n+1} - b_n < 0. Multiplying the inequality by this negative difference (and reversing the inequality direction) yields

(L - \frac{\varepsilon}{2})(b_{n+1} - b_n) > a_{n+1} - a_n > (L + \frac{\varepsilon}{2})(b_{n+1} - b_n)

for all n > N. Summing these inequalities from n = k to m-1 where k > N and m > k gives

(L - \frac{\varepsilon}{2})(b_m - b_k) > a_m - a_k > (L + \frac{\varepsilon}{2})(b_m - b_k).

Now, let m \to \infty. Since a_m \to 0 and b_m \to 0, this simplifies to

(L - \frac{\varepsilon}{2})(- b_k) > - a_k > (L + \frac{\varepsilon}{2})(- b_k).

Multiplying through by -1 (and reversing the inequalities) yields

(L - \frac{\varepsilon}{2}) b_k < a_k < (L + \frac{\varepsilon}{2}) b_k.

Dividing by b_k > 0 (preserving inequalities) results in

L - \frac{\varepsilon}{2} < \frac{a_k}{b_k} < L + \frac{\varepsilon}{2}.

Since \varepsilon > 0 is arbitrary and k > N can be arbitrarily large, \lim_{k \to \infty} \frac{a_k}{b_k} = L. This preserves the monotonicity of b_n in the differences and ensures the limit transfers under the transformation of summing backwards to the limit point at infinity. The case L = \pm \infty follows analogously by showing the ratio exceeds or falls below arbitrary thresholds for large n.^[8] For the special handling when b_n > 0 is decreasing, the sign adjustment in the differences (as b_{n+1} - b_n < 0) is crucial to maintain the inequality directions during summation, ensuring the bounds align with the convergence to zero. This approach verifies the conditions of monotonicity preservation and limit existence directly on the original sequences without needing full reversal, though it mirrors the \infty/\infty proof by effectively transforming the tail into a divergent framework via the limits at infinity.^[8]

Applications and Examples

Arithmetic and Geometric Means

One classical application of the Stolz–Cesàro theorem arises in establishing the convergence of the arithmetic mean of a sequence. Suppose \{a_n\} is a sequence of real numbers that converges to a limit L \in \mathbb{R}. Then the arithmetic mean \frac{1}{n} \sum_{k=1}^n a_k also converges to L. To see this, let b_n = \sum_{k=1}^n a_k and consider the ratio b_n / n. The sequence \{n\} is strictly increasing and unbounded. Applying the Stolz–Cesàro theorem in the \infty/\infty form yields

\lim_{n \to \infty} \frac{b_n}{n} = \lim_{n \to \infty} \frac{b_{n+1} - b_n}{ (n+1) - n } = \lim_{n \to \infty} a_{n+1} = L,

provided the latter limit exists.^[1]^[3] A related representation expresses each term as a_n = a_1 + \sum_{k=1}^{n-1} (a_{k+1} - a_k), which allows the sum \sum_{k=1}^n a_k to be rewritten in terms of these differences. Under the convergence of \{a_n\} to L, the differences a_{k+1} - a_k converge to 0, and applying the theorem with denominator sequence b_n = n confirms that the average preserves the limit L. This approach highlights the theorem's role in handling averages of sequences where direct summation might obscure the limiting behavior.^[1] For the geometric mean, consider a positive sequence \{a_n\} with a_n > 0 for all n and \lim_{n \to \infty} a_n = L > 0. The geometric mean \left( \prod_{k=1}^n a_k \right)^{1/n} then also converges to L. To prove this, apply the natural logarithm to obtain

\frac{1}{n} \sum_{k=1}^n \log a_k = \log \left( \left( \prod_{k=1}^n a_k \right)^{1/n} \right).

Let c_n = \sum_{k=1}^n \log a_k. By the arithmetic mean result above (or directly via Stolz–Cesàro), \lim_{n \to \infty} c_n / n = \lim_{n \to \infty} \log a_n = \log L, since \{\log a_n\} converges. Exponentiating both sides, the geometric mean limit follows by the continuity of the exponential function. The positivity condition ensures the logarithms are defined and the means are well-posed.^[1] These applications demonstrate the Stolz–Cesàro theorem's utility in preserving limits under averaging operations, serving as foundational tools in sequence analysis for both additive and multiplicative structures.^[1]

Specific Limit Computations

The Stolz–Cesàro theorem is particularly useful for evaluating limits of sequences that result in indeterminate forms like ∞/∞, where direct computation is intractable due to the cumulative nature of the sequences involved. One such example is the limit \lim_{n \to \infty} \frac{H_n}{\log n} = 1, where H_n = \sum_{k=1}^n \frac{1}{k} is the n-th harmonic number. Both H_n and \log n tend to infinity as n → ∞, and \log n is strictly increasing. Applying the ∞/∞ case of the theorem with a_n = H_n and b_n = \log n, the difference quotient is

\frac{a_{n+1} - a_n}{b_{n+1} - b_n} = \frac{\frac{1}{n+1}}{\log(n+1) - \log n} = \frac{1/(n+1)}{\log(1 + 1/n)}.

Since \log(1 + 1/n) \sim 1/n as n → ∞ and 1/(n+1) \sim 1/n, the quotient tends to 1. Thus, the theorem implies the desired limit holds. This result aligns with the asymptotic expansion H_n \sim \log n + \gamma, where \gamma is the Euler-Mascheroni constant, confirming the leading term behavior.^[8] Another illustrative example is the limit \lim_{n \to \infty} n^{1/n} = 1. Direct computation of the root is challenging, but taking the natural logarithm yields \log(n^{1/n}) = (\log n)/n, an ∞/∞ form. Setting a_n = \log n and b_n = n, both tend to infinity, and b_n is strictly increasing. The difference quotient simplifies to

\frac{a_{n+1} - a_n}{b_{n+1} - b_n} = \log\left(1 + \frac{1}{n}\right) / 1 \to 0

as n → ∞. By the theorem, \lim_{n \to \infty} (\log n)/n = 0, so exponentiating gives the original limit. This demonstrates how the theorem reduces the problem to a simpler continuous limit via differences.^[8] For limits involving factorials, consider \lim_{n \to \infty} \frac{(n!)^{1/n}}{n} = \frac{1}{e}. Direct evaluation is difficult, but analyzing the logarithm leads to \lim_{n \to \infty} \frac{\log n!}{n \log n} = 1 as the leading asymptotic term. Here, a_n = \log n! = \sum_{k=1}^n \log k and b_n = n \log n, both tending to infinity, with b_n strictly increasing for large n. The difference quotient is

\frac{a_{n+1} - a_n}{b_{n+1} - b_n} = \frac{\log(n+1)}{(n+1)\log(n+1) - n \log n} = \frac{\log(n+1)}{\log(n+1) + n \log(1 + 1/n)}.

As n → ∞, n \log(1 + 1/n) \to 1 and \log(n+1) \sim \log n \to \infty, so the quotient behaves as \log n / (\log n + 1) \to 1. The theorem thus establishes the limit 1 for \log n! / (n \log n). The finer 1/e behavior arises from the next-order term in Stirling's approximation, \log n! \sim n \log n - n + \frac{1}{2} \log(2\pi n), implicitly supported by the theorem's leading confirmation.^[8]

General Form

Statement

The general form of the Stolz–Cesàro theorem addresses indeterminate forms of type \infty/\infty or $0/0 for sequences \{a_n\} and \{b_n\}, where the limit of a_n/b_n may not exist, by employing liminf and limsup to capture partial convergence behaviors, particularly useful for oscillatory sequences.^[9] Assume \{b_n\} is a strictly increasing sequence with b_n \to +\infty, and let \Delta a_n = a_{n+1} - a_n, \Delta b_n = b_{n+1} - b_n denote the forward differences. Then,

\liminf_{n \to \infty} \frac{\Delta a_n}{\Delta b_n} \le \liminf_{n \to \infty} \frac{a_n}{b_n} \le \limsup_{n \to \infty} \frac{a_n}{b_n} \le \limsup_{n \to \infty} \frac{\Delta a_n}{\Delta b_n},

provided the expressions are well-defined (noting that \Delta b_n > 0 for all sufficiently large n due to strict monotonicity).^[9] If \lim_{n \to \infty} \frac{\Delta a_n}{\Delta b_n} = L exists (finite or \pm \infty), then \lim_{n \to \infty} \frac{a_n}{b_n} = L. This general statement encompasses the basic \infty/\infty and $0/0 cases as special instances where the difference limit exists.^[9]

Proof

The general form of the Stolz–Cesàro theorem states that if (b_n) is a strictly increasing sequence of real numbers with b_n \to \infty as n \to \infty, and (a_n) is any real sequence, then

\liminf_{n \to \infty} \frac{\Delta a_n}{\Delta b_n} \leq \liminf_{n \to \infty} \frac{a_n}{b_n} \leq \limsup_{n \to \infty} \frac{a_n}{b_n} \leq \limsup_{n \to \infty} \frac{\Delta a_n}{\Delta b_n},

where \Delta a_n = a_{n+1} - a_n and \Delta b_n = b_{n+1} - b_n > 0.^[1] To prove the liminf inequality, let \ell = \liminf_{n \to \infty} \frac{\Delta a_n}{\Delta b_n}. If \ell = +\infty, the inequality holds trivially since the left side is infinite. Assume \ell is finite. For any \varepsilon > 0, there exists N \in \mathbb{N} such that for all n \geq N,

\frac{\Delta a_n}{\Delta b_n} > \ell - \varepsilon.

Fix n \geq N and consider m > n. Then,

a_m - a_n = \sum_{k=n}^{m-1} \Delta a_k > (\ell - \varepsilon) \sum_{k=n}^{m-1} \Delta b_k = (\ell - \varepsilon)(b_m - b_n),

a_m > a_n + (\ell - \varepsilon)(b_m - b_n).

Dividing by b_m > 0,

\frac{a_m}{b_m} > (\ell - \varepsilon) + \frac{a_n - (\ell - \varepsilon) b_n}{b_m}.

As m \to \infty, the second term tends to 0 because b_m \to \infty. Thus, for sufficiently large m > n,

\frac{a_m}{b_m} > \ell - \varepsilon,

which implies

\liminf_{m \to \infty} \frac{a_m}{b_m} \geq \ell - \varepsilon.

Since \varepsilon > 0 is arbitrary,

\liminf_{n \to \infty} \frac{a_n}{b_n} \geq \ell = \liminf_{n \to \infty} \frac{\Delta a_n}{\Delta b_n}.[1]

The limsup inequality is proved analogously. Let L = \limsup_{n \to \infty} \frac{\Delta a_n}{\Delta b_n}. If L = -\infty, the inequality holds trivially. Assume L is finite. For any \varepsilon > 0, there exists N \in \mathbb{N} such that for all n \geq N,

\frac{\Delta a_n}{\Delta b_n} < L + \varepsilon.

Fix n \geq N and m > n. Then,

a_m - a_n = \sum_{k=n}^{m-1} \Delta a_k < (L + \varepsilon)(b_m - b_n),

a_m < a_n + (L + \varepsilon)(b_m - b_n).

Dividing by b_m,

\frac{a_m}{b_m} < (L + \varepsilon) + \frac{a_n - (L + \varepsilon) b_n}{b_m}.

As m \to \infty, the second term tends to 0, so for sufficiently large m > n,

\frac{a_m}{b_m} < L + \varepsilon,

implying

\limsup_{m \to \infty} \frac{a_m}{b_m} \leq L + \varepsilon.

Since \varepsilon > 0 is arbitrary,

\limsup_{n \to \infty} \frac{a_n}{b_n} \leq L = \limsup_{n \to \infty} \frac{\Delta a_n}{\Delta b_n}.[1]

A key insight underlying these proofs is that for m > n,

\frac{a_m - a_n}{b_m - b_n} = \sum_{k=n}^{m-1} \frac{\Delta a_k}{\Delta b_k} \cdot w_k,

where w_k = \frac{\Delta b_k}{b_m - b_n} \geq 0 are weights summing to 1, forming a convex combination of the ratios \frac{\Delta a_k}{\Delta b_k} for k = n, \dots, m-1. Since (b_n) is strictly increasing, the monotonicity ensures that if the ratios \frac{\Delta a_k}{\Delta b_k} are eventually bounded below by \ell - \varepsilon (resp. above by L + \varepsilon), the weighted average is similarly bounded, and taking limits propagates the infimum (resp. supremum) bounds to the ratios \frac{a_m}{b_m}. This weighted average property holds in the infinite case as b_n \to \infty allows the fixed initial terms to vanish in the limit.^[1] For the case where (b_n) is non-strictly increasing (i.e., non-decreasing with b_n \to \infty), the result extends by a density argument: consider the subsequence of indices where b_n strictly increases, which is cofinal in \mathbb{N} since b_n \to \infty; the liminf and limsup of \frac{a_n}{b_n} and \frac{\Delta a_n}{\Delta b_n} (defined only where \Delta b_n > 0) coincide along this subsequence, preserving the inequalities. Alternatively, a small perturbation can make b_n strictly increasing without altering the limits.^[1]

Relations and Extensions

Analogy with L'Hôpital's Rule

The Stolz–Cesàro theorem is widely regarded as the discrete analogue of L'Hôpital's rule, providing a method to evaluate limits of quotients of sequences that yield indeterminate forms of type \infty/\infty or $0/0 by replacing the original sequences with their forward differences.^[10]^[11] In particular, while L'Hôpital's rule applies to functions f and g where \lim_{x \to a} \frac{f(x)}{g(x)} is indeterminate and the functions are differentiable near a with g'(x) \neq 0, the limit equals \lim_{x \to a} \frac{f'(x)}{g'(x)} if the latter exists. Analogously, for sequences \{a_n\} and \{b_n\}, if \{b_n\} is strictly monotonic and unbounded (typically increasing to \infty) and \lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L exists, then \lim_{n \to \infty} \frac{a_n}{b_n} = L.^[10] A key structural difference lies in the conditions imposed: L'Hôpital's rule relies on local differentiability of the functions involved, whereas the Stolz–Cesàro theorem demands global monotonicity and divergence of the denominator sequence \{b_n\}, reflecting the inherent constraints of discrete structures.^[10] This makes the theorem particularly suited to non-differentiable discrete objects, such as combinatorial sequences or partial sums, where continuous approximation via derivatives is unavailable.^[11] Repeated applications of the Stolz–Cesàro theorem mirror the higher-order iterations of L'Hôpital's rule, allowing resolution of persistent indeterminate forms by successively differencing the sequences until a determinate limit emerges.^[10] This iterative process preserves the analogy while adapting to the discrete setting, though it requires verifying the monotonicity condition at each step.

Broader Connections and Generalizations

The Stolz–Cesàro theorem finds significant connections to Tauberian theory, particularly as a foundational tool in the Hardy–Littlewood framework for analyzing power series and Dirichlet series with positive coefficients. In this context, the theorem underpins results where Cesàro summability of partial sums implies ordinary convergence, provided Tauberian conditions like non-negativity or bounded oscillation are satisfied; for instance, Hardy and Littlewood's 1913 work establishes such implications by leveraging discrete limit criteria akin to Stolz–Cesàro for asymptotic behavior of coefficients.^[12] These links highlight the theorem's role in bridging summability methods, where it serves as a discrete analog facilitating proofs of Tauberian converses for series expansion.^[13] In modern applications, the theorem appears in stochastic processes, notably renewal theory, where it computes limits of renewal rates and expected costs in processes with alternating geometric interarrivals, ensuring convergence of time averages to ensemble averages.^[14] For instance, in partial resetting models of diffusion processes, Stolz–Cesàro derives time-dependent densities by evaluating asymptotic ratios of renewal-like sums.^[15] The converse of the Stolz–Cesàro theorem, implying the difference quotient limit from the overall quotient limit, holds under supplementary Tauberian conditions, such as slow oscillation of the sequences or bounded increments n(u_n - u_{n-1}) = O(1), ensuring the implication reverses without monotonicity alone.^[16] These conditions align with weakly vanishing mean oscillation properties in Tauberian theory, providing necessary and sufficient criteria for summability inversions.^[17]

Historical Development

Origins

The Stolz–Cesàro theorem was first introduced by the Austrian mathematician Otto Stolz in his 1879 paper "Über die Grenzwerte der Quotienten" in Mathematische Annalen (vol. 15, pp. 556–559), where he stated and proved the ∞/∞ case as a method for evaluating limits of quotients of sequences, particularly those arising from divergent series.^[18] Stolz later included a proof in his 1885 monograph Vorlesungen über allgemeine Arithmetik nach den neueren Ansichten (pp. 173–175), within a broader discussion of sequence limits and arithmetic operations on real numbers, emphasizing its utility in handling indeterminate forms where both numerator and denominator tend to infinity, a common challenge in analyzing partial sums of series.^[19] Independently, the Italian mathematician Ernesto Cesàro proved the theorem in his 1888 paper "Sur la multiplication des séries" published in Atti della Reale Accademia delle Scienze di Torino (vol. 23, p. 54).^[20] Cesàro's contribution addressed limits of quotients in the context of series multiplication, providing a complementary perspective for limit computations in analysis. There is no documented evidence of prior work on this specific criterion, and the theorem is accordingly attributed to Stolz and Cesàro as original discoveries. These publications occurred amid the late 19th-century surge in rigorous real analysis, driven by efforts to resolve issues in infinite series and convergence following Bernhard Riemann's foundational investigations into sums and integrals around 1850–1870. Stolz's emphasis on the ∞/∞ form directly addressed motivations from divergent series, reflecting the era's need for discrete analogs to continuous limit techniques in the wake of Weierstrass's influence on epsilon-delta rigor.^[19]

Later Contributions

The Stolz–Cesàro theorem gained significant popularity following its inclusion as Problem 70 in George Pólya and Gábor Szegő's influential 1925 volume Aufgaben und Lehrsätze aus der Analysis, which presented it as a key tool for sequence limits and helped integrate it into advanced analysis education. This exposure in a widely used problem collection contributed to its dissemination beyond the original publications by Otto Stolz and Ernesto Cesàro. In the mid-20th century, refinements extended the theorem's scope, notably Konrad Knopp's 1927 generalization to cases involving liminf and limsup in his treatise Theorie und Anwendung der unendlichen Reihen, where it was applied to infinite series convergence under weaker limit assumptions. The theorem's recognition in Tauberian theory emerged during the 1930s, with connections to Fourier analysis through Norbert Wiener's work on summation methods and their inverses, bridging discrete limits to continuous transforms. By the 1950s, the general form of the theorem—applicable to broader sequence behaviors—was prominently featured in probability literature, such as William Feller's An Introduction to Probability Theory and Its Applications (1950), where it supported analyses of stochastic processes and renewal theory. The theorem continued to appear in standard analysis textbooks, including Tom M. Apostol's Mathematical Analysis (1974), which discusses its role in limit evaluations for quotients of sequences. In more recent computational contexts, particularly within discrete calculus during the 2010s, it has been employed to derive closed forms for sums, such as those of integer powers, by treating partial sums as sequences amenable to the theorem's discrete analogue of L'Hôpital's rule.^[21] These applications underscore its enduring utility in bridging classical analysis with modern numerical methods.

References

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[PDF] The Stolz-Cesaro Theorem - KSU Math
The Stolz-Cesaro Theorem has numerous applications in Calculus. Below are three of the most significant ones. ”Additive” Cesaro's Theorem.
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[PDF] Exact Asymptotics for Linear Quadratic Adaptive Control
Stolz–Cesàro theorem: Theorem H.1 (Stolz–Cesàro). Let {at}t≥1 and {bt}t≥1 be two sequences of real numbers. Assume that. {bt}t≥1 is a strictly monotone ...<|control11|><|separator|>
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[PDF] Research Paper Article on the Application of Stolz–Cesàro Theorem ...
Dec 31, 2023 · Similar to L' hospital rule we have a result for sequences and we called it as Stolz Cesàro result. Stolz–Cesàro theorem is named after Otto ...
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[PDF] L'hôpital's Monotone rule, Gromov's theorem, and operations that ...
Jan 1, 2017 · We shall now give a proof of Gromov's theorem by also employing a change of variables. We start with the following very simple special case.
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[PDF] Analysis
Oct 7, 2009 · Cesaro-Stolz Theorem. Let {xn} and {yn} be two sequences of real numbers, where the yn are positive, strictly increasing, and unbounded. If lim.
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https://doi.org/10.2298/PIM1715011E
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[PDF] Plausible and genuine extensions of L'Hospital's Rule
We do not know who coined the name of this very well known theorem. The ∞/∞ case is stated and proved on pages 173–175 of Stolz's 1885 book [4] and also ...
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https://doi.org/10.1007/978-3-030-79431-6
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The Ali-Cesaro Stolz Theorem: Extending Classical Limit Analysis ...
Aug 13, 2024 · This novel theorem leverages the properties of limits and Z-transforms, offering a robust framework for analyzing sequences and series.
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[PDF] Hyers-Ulam stability of the first order difference equation generated ...
Oct 2, 2022 · The following Stolz-Cesàro theorem [14] is about the inequality of limit. Lemma 2.1 (Stolz-Cesàro theorem). ... j=1 tjKj ≥ lim inf n ...
[11]
Stolz–Cesàro theorem | Brilliant Math & Science Wiki
Stolz–Cesàro theorem is a powerful tool for evaluating limits of sequences, and it is a ... ∞/∞ case): Let ( a n ) n ∈ N (a_n)_{n\in\mathbb N} (an) ...Missing: statement | Show results with:statement
[12]
Plausible and genuine extensions of L'Hospital's Rule - ResearchGate
For limits, the discrete analogue of L'Hospital's Rule is the Stolz-Cesàro Theorem. ... generating the indeterminate form 0/0 at inﬁnity with bnstrictly ...
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[PDF] the general theory of dirichlet's series - JScholarship
This general idea may be made more precise by the following theorem, which includes Theorem 17 as a special case, and may be established by reasoning of the ...Missing: statement | Show results with:statement
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Aug 9, 2025 · Generally speaking, by following the pattern of Hardy, theorems about the relationship between the asymptotics of Cesaro and Abel means are ...
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[PDF] Nets and Filters
We will then show that nets can be used interchangeably with filters, another generalization of sequences in a topological space. 1 Where sequences fail. 1.1 ...
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(PDF) On Cesàro summability of vector valued multiplier spaces and ...
Aug 7, 2025 · In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between ...
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(PDF) Weighted and Higher Order Cesàro Means in Banach Spaces ...
Nov 23, 2023 · PDF | On Nov 23, 2023, Luigi Accardi Centro and others published Weighted and Higher Order Cesàro Means in Banach Spaces and Applications to ...
[19]
[PDF] Warranty Cost Analysis with an Alternating Geometric Process - arXiv
It can be shown that the series (12) is divergent (based on d'Alembert's test and the Stolz-Cesàro theorem [15]). Hence, 𝐸(𝐶(𝑊𝑇)) goes to infinity. Therefore, ...
[20]
Time-dependent probability density function for partial resetting ...
Aug 4, 2023 · The derivation uses Stolz-Cesaro's theorem [95] ... this model correspond to total resetting [34] and a stochastic process without resetting,.
[21]
[PDF] Regularly Varying Functions and Power Series Methods - CORE
Using a variant of a theorem of Stolz (cf. Adamovid [l; Lemme I]) we get by. P ... We start this section with a Tauberian theorem for the MD-method. THEOREM.
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The classical Cesaro's theorems apply to ratios of real-valued sequences like f, g, where g, – c monotonically, or f, – 0, and g, – 0 monotonically.
[23]
A criterion for the limit of a ratio of functions | Request PDF
... Tauberian converse of the Stolz–Cesàro theorem. In this paper, we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity ...
[24]
[PDF] Necessary and sufficient Tauberian condition for both Cesàro ... - HAL
Oct 4, 2023 · Moreover, Lemma 2.4 provides an alternative simple proof of Hardy-Littlewood (1914) positive Tauberian theorem. Let us observe that the ...
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Vorlesungen über allgemeine Arithmetik; nach den neueren Ansichten
Dec 3, 2008 · Vorlesungen über allgemeine Arithmetik; nach den neueren Ansichten. by: Stolz, Otto, 1881-. Publication date: 1885. Topics: Algebra, Arithmetic.
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Otto Stolz (1842 - 1905) - Biography - MacTutor History of Mathematics
In 1885 the first part of Vorlesungen über Allgemeine Arithmetik T. (Lectures on general arithmetic). was published on Allgemeines und Arithhmetik der Reelen ...
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Ernesto Cesàro (1859 - 1906) - Biography - MacTutor
Cesàro's main contribution was to differential geometry. Influenced by Darboux while in Paris he formulated 'intrinsic geometry'.
[28]
Sums of Integer Powers via the Stolz-Cesàro Theorem - ResearchGate
Aug 10, 2025 · Abstract. The Stoltz-Cesàro Theorem, a discrete version of l'Hôpital's rule, is applied to the summation of integer powers.