convergent series

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1: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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2: 36.15 Methods of Computation

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§36.15(i) Convergent Series

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3: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►Absolutely convergent series are also convergent. … ►

Weierstrass $M$-test

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§1.9(vi) Power Series

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4: 3.9 Acceleration of Convergence

§3.9 Acceleration of Convergence

… ►Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums. … ►If $S={\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k$ is a convergent series, then … ►Examples are provided by the following analytic transformations of slowly-convergent series into rapidly convergent ones: … ►For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).

5: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …

6: 28.34 Methods of Computation

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Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

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7: 12.4 Power-Series Expansions

… ►These series converge for all values of $z$.

8: 27.5 Inversion Formulas

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27.5.7 $$G(x)=\sum _{m=1}^{\mathrm{\infty }}\frac{F(mx)}{m^s}⟺F(x)=\sum _{m=1}^{\mathrm{\infty }}\mu \left(m\right)\frac{G(mx)}{m^s},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $m$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

Referenced by:

§27.17, §27.5

Permalink:

http://dlmf.nist.gov/27.5.E7

Encodings:

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See also:

Annotations for §27.5 and Ch.27

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9: 14.28 Sums

… ►The series converges uniformly for $z_1$ outside or on ${ℰ}_1$, and $z_2$ within or on ${ℰ}_2$. …

10: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for . … ►However, by appropriate choice of the constant $z_0$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. …