I am trying to evaluate the infinite series:
$$\sum_{k=1}^{\infty} \frac{1}{k(k+c)} $$
where c is a constant, positive integer.
Is my approach correct by using partial fraction decomposition? Then I break down the series into the form $\frac{1/c}{k} - \frac{1/c}{k+c}$ and I get a series of the form (after writing out few terms and noticing some cancellations)
$$\frac{1/c}{1} + \frac{1/c}{2} + \frac{1/c}{3} + ...$$
Is there a way for me to formalize the solution better (if correct), maybe into some kind of compact formula? Otherwise is there a correct/better approach to the problem?
I also used an online infinite series calculator which told me the sum of the series for different values of c. For c = 1,2,3,4 the sum of the infinite series evaluates to $1, \frac{3}{4}, \frac{11}{18}, \frac{25}{48}$ respectively. However, I don't really see the pattern here.
Any helps/hints would be much appreciated! Thank you!
$$\sum_{n\geq 1}\frac{1}{n(n+m)}=\frac{1}{m}\sum_{n\geq 1}\left(\frac{1}{n}-\frac{1}{n+m}\right)=\frac{1}{m}\sum_{n=1}^{m}\frac{1}{n}=\frac{H_{m}}{m} $$ where $H_s$ is the $s$-th harmonic number.