convergent series

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21: 19.27 Asymptotic Approximations and Expansions

… ►These series converge but not fast enough, given the complicated nature of their terms, to be very useful in practice. …

22: 21.2 Definitions

… ►This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $𝐳$ and $𝛀$ spaces; hence $\theta \left(𝐳\right|𝛀)$ is an analytic function of (each element of) $𝐳$ and (each element of) $𝛀$. …

23: 6.16 Mathematical Applications

… ►Compare Figure 6.16.1. …

24: 28.11 Expansions in Series of Mathieu Functions

… ►The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip $S$. …

25: 1.15 Summability Methods

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§1.15(ii) Regularity

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26: 2.11 Remainder Terms; Stokes Phenomenon

… ►Even when the series converges this is unwise: the tail needs to be majorized rigorously before the result can be guaranteed. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. …

27: 27.14 Unrestricted Partitions

… ►Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for $p\left(n\right)$: …

28: 1.10 Functions of a Complex Variable

… ►The right-hand side is the Taylor series for $f(z)$ at $z=z_0$, and its radius of convergence is at least $R$. … ►The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. … ►where $\mu >0$, $f_0\ne 0$, and the series converges in a neighborhood of $z_0$. … ►Let $F(x,z)$ have a converging power series expansion of the form …

29: 19.36 Methods of Computation

… ►The incomplete integrals $R_F(x,y,z)$ and $R_G(x,y,z)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_C$, accompanied by two quadratically convergent series in the case of $R_G$; compare Carlson (1965, §§5,6). … ►Faster convergence of power series for $K\left(k\right)$ and $E\left(k\right)$ can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …

30: 4.13 Lambert $W$-Function

… ►For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. In the case of $k=0$ and real $z$ the series converges for $z\ge \mathrm{e}$. …