<s>[INST] How to find the sum of this infinite series. How to find the sum of the following series ?
Kindly guide me about the general term, then I can give a shot at summing it up.
$$1 - \frac{1}{4} + \frac{1}{6} -\frac{1}{9} +\frac{1}{11} - \frac{1}{14} + \cdots$$
 [/INST] Using the principal value for the doubly infinite harmonic series yields
$$
\begin{align}
\sum_{k=0}^{\infty} \left(\dfrac1{5k+1} - \dfrac1{5k+4}\right)
&=\frac15\sum_{k=-\infty}^\infty\frac{1}{k+1/5}\\
&=\frac{\pi}{5}\cot\left(\frac{\pi}{5}\right)\\
&=\frac{\pi}{5}\sqrt{\frac{5+2\sqrt{5}}{5}}
\end{align}
$$
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