Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Question

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$

For $a,b$ integer we can use Partial fraction decomposition. Using Maple for the general case it seems the 'closed' form is $$ \frac{\Psi(1+a)-\Psi(1+b)}{a-b} $$

Any ideas how to show that?

We can use partial fraction decomposition even when $a$ and $b$ aren't integers. — Joshua Mundinger, Aug 15 '14 at 18:53

score 7 · Accepted Answer · edited Aug 16 '14 at 01:59

7

$$\begin{eqnarray*}\psi(1+a)-\psi(1+b)&=&\sum_{n=1}^{\infty}\left(\frac{a}{n(n+a)}-\frac{b}{n(n+b)}\right)=\sum_{n=1}^{\infty}\frac{a(n+b)-b(n+a)}{n(n+a)(n+b)}\\&=&(a-b)\sum_{n=1}^{\infty}\frac{1}{(n+a)(n+b)}.\end{eqnarray*}$$

edited Aug 16 '14 at 01:59

Jack D'Aurizio

353,855

answered Aug 15 '14 at 18:35

user71352

13,038

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

1 Answers1