# infinite series

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## 1—10 of 66 matching pages

##### 1: 4.11 Sums
For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).
##### 2: 25.8 Sums
25.8.1 $\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1\right)=1.$
25.8.2 $\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(k+1)!}\left(\zeta\left(s+k% \right)-1\right)=\Gamma\left(s-1\right),$ $s\neq 1,0,-1,-2,\dots$.
25.8.5 $\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}=-\gamma z-z\psi\left(1-z\right),$ $|z|<1$.
25.8.9 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac% {1}{2}\ln 2.$
25.8.10 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}% -\frac{7}{4\pi^{2}}\zeta\left(3\right).$
##### 3: 24.8 Series Expansions
###### §24.8(i) Fourier Series
If $n=1,2,\dots$ and $0\leq x\leq 1$, then …
###### §24.8(ii) Other Series
24.8.9 $E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$ $n=1,2,\dots$.
##### 4: 25.2 Definition and Expansions
25.2.1 $\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$
###### §25.2(ii) Other InfiniteSeries
25.2.2 $\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}},$ $\Re s>1$.
25.2.3 $\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^% {s}},$ $\Re s>0$.
25.2.4 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
##### 6: 15.15 Sums
For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …
##### 7: 25.16 Mathematical Applications
25.16.1 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$
25.16.5 $H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},$
25.16.14 $\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}=\frac{5}{4}\zeta\left(3% \right),$
25.16.15 $\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}\zeta\left(3% \right).$
##### 8: 16.11 Asymptotic Expansions
For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows:
16.11.1 $E_{p,q}(z)=(2\pi)^{\ifrac{(p-q)}{2}}\kappa^{-\nu-(\ifrac{1}{2})}{\mathrm{e}}^{% \kappa z^{\ifrac{1}{\kappa}}}\sum_{k=0}^{\infty}c_{k}\left(\kappa z^{\ifrac{1}% {\kappa}}\right)^{\nu-k},$ $p,
16.11.2 $H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\Gamma\left(a_{% m}+k\right)\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\Gamma\left(a_{\ell}-a_{m}-k\right)}{\prod\limits% _{\ell=1}^{q}\Gamma\left(b_{\ell}-a_{m}-k\right)}}\right)z^{-a_{m}-k}.$
16.11.7 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).$
16.11.8 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),$
##### 9: 8.15 Sums
8.15.2 $a\sum_{k=1}^{\infty}\left(\frac{{\mathrm{e}}^{2\pi\mathrm{i}k(z+h)}}{\left(2% \pi\mathrm{i}k\right)^{a+1}}\Gamma\left(a,2\pi\mathrm{i}kz\right)+\frac{{% \mathrm{e}}^{-2\pi\mathrm{i}k(z+h)}}{\left(-2\pi\mathrm{i}k\right)^{a+1}}% \Gamma\left(a,-2\pi\mathrm{i}kz\right)\right)=\zeta\left(-a,z+h\right)+\frac{z% ^{a+1}}{a+1}+\left(h-\tfrac{1}{2}\right)z^{a},$ $h\in[0,1]$.
For other infinite series whose terms include incomplete gamma functions, see Nemes (2017a), Reynolds and Stauffer (2021), and Prudnikov et al. (1986b, §5.2).
##### 10: 25.12 Polylogarithms
25.12.8 $\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}=\frac{\pi^{2}}{6}-% \frac{\pi\theta}{2}+\frac{\theta^{2}}{4}.$
25.12.9 $\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}% \ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\mathrm{d}x.$
25.12.10 $\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}.$