# infinite

(0.001 seconds)

## 1—10 of 115 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).
##### 4: 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 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}% -\frac{7}{4\pi^{2}}\zeta\left(3\right).$
##### 5: 25.2 Definition and Expansions
25.2.1 $\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$
###### §25.2(ii) Other Infinite Series
25.2.4 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
###### §25.2(iv) Infinite Products
25.2.11 $\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1},$ $\Re s>1$,
##### 6: Sidebar 5.SB1: Gamma & Digamma Phase Plots
This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. …
##### 8: 1.9 Calculus of a Complex Variable
###### §1.9(v) Infinite Sequences and Series
This sequence converges pointwise to a function $f(z)$ if …
##### 9: 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 $E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$ $n=1,2,\dots$.