Find the sum of series $\sum_{k=1}^\infty \frac{1}{k^2+2k}$

Question

Find the sum of the series.$$\sum_{k=1}^\infty \frac{1}{k^2+2k}$$

Which technique should I use? I tried but I cannot find anything.

Answer 1

Here is another method using power series and double integrals. Let $$ f(x)=\sum_{k=1}^\infty \frac{1}{k^2+2k}x^{k+2} $$ and then $$ f(0)=0,f'(0)=0, f'(x)=\sum_{k=1}^\infty \frac{1}{k}x^{k+1}, \bigg(\frac{f'(x)}{x}\bigg)'=\sum_{k=1}^\infty x^{k-1}=\frac{1}{1-x}. $$ So \begin{eqnarray} f(1)&=&\int_0^1\int_0^x\frac{x}{1-t}dtdx=\int_0^1\int_t^1\frac{x}{1-t}dxdt\\ &=&\int_0^1\frac{1}{1-t}\cdot\frac12(1-t^2)dt=\frac12\int_0^1(1+t)dt\\ &=&\frac34. \end{eqnarray}