convergent series

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11: 1.8 Fourier Series

… ►

§1.8(ii) Convergence

… ►Then the series (1.8.1) converges to the sum … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …

12: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►The series (28.19.2) converges absolutely and uniformly on compact subsets within $S$. …

13: 14.13 Trigonometric Expansions

… ►These Fourier series converge absolutely when . …

14: 16.15 Integral Representations and Integrals

… ►These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. …

15: 19.12 Asymptotic Approximations

… ►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: …

16: 17.4 Basic Hypergeometric Functions

… ►The infinite series converges for all $z$ when $s>r$, and for when $s=r$. … ►The infinite series converge when $s\ge r$ provided that and also, in the case $s=r$, . …

17: 27.7 Lambert Series as Generating Functions

… ►If , then the quotient $x^n/(1-x^n)$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: …

18: 26.8 Set Partitions: Stirling Numbers

… ►when $f(x)$ is analytic for all $x$, and the series converges, where … ►when $f(x)$ is analytic for all $x$, and the series converges. …

19: 17.18 Methods of Computation

… ►Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. …

20: 2.1 Definitions and Elementary Properties

… ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, … ►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. …