Evaluating series using digamma

Evaluate the series:

$$\sum _{n=0}^{\infty }\frac{1}{(4n+1)(4n+3)}$$

Solution



Successively we have that:

$$\begin{aligned}
\sum_{n=0}^{\infty}\frac{1}{\left ( 4n+1 \right )\left ( 4n+3 \right )} &= \frac{1}{2}\sum_{n=0}^{\infty}\left [ \frac{1}{4n+1}- \frac{1}{4n+3} \right ]\\
 &= \frac{1}{8}\sum_{n=0}^{\infty}\left [ \frac{1}{n+1/4}- \frac{1}{n+3/4} \right ]\\
 &= \frac{1}{8}\left [ \psi \left ( \frac{3}{4} \right )- \psi \left ( \frac{1}{4} \right ) \right ]\\
 &= \frac{\pi\cot \frac{\pi}{4}}{8}= \frac{\pi}{8}
\end{aligned}$$

We used the known formula $\psi\left ( 1-z \right )-\psi(z)= \pi \cot \pi z$.