Calculer les sommes des séries suivantes :
1
\(\sum_{k=2}^\infty \frac 1{k^2-1}\).
2
\(\sum_{k=1}^\infty \frac 1{k(k+1)(k+2)}\).
3
\(\sum_{k=1}^\infty \frac 1{k(k+1)\dots(k+p)}\).
4
\(\sum_{k=0}^\infty \frac 1{k^3+8k^2+17k+10}\).
5
\(\sum_{k=1}^\infty \ln\left(1+\frac2{k(k+3)}\right)\).
6
\(\sum_{k=2}^\infty \ln\left(1-\frac1{k^2}\right)\).
7
\(\sum_{k=0}^\infty \ln\left(\cos\frac\alpha{2^k}\right)\).
8
\(\sum_{k=0}^\infty 2^{-k}\tan(2^{-k}\alpha)\).
9
\(\sum_{k=0}^\infty \frac{2k^3-3k^2+1}{(k+3)!}\).
10
\(\sum_{n=p}^\infty C_n^p x^n\).
11
\(\sum_{k=1}^\infty \frac{x^k}{(1-x^k)(1-x^{k+1})}\).
12
\(\sum_{k=1}^\infty \frac{k-n[k/n]}{k(k+1)}\).