Prove $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\cdots$ converges. Consider the series:
$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\cdots$ 
Is it convergent?
I believe I need to find a way to split the terms into additive and subtraction terms, however I'm not sure how to do this and as a result prove the outcome. 

\begin{align}
\dots &= \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots \right) - \left(\frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \dots \right) \\
&= \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots \right)- \frac{1}{2} \, \left(1 - \frac{1}{2} + \frac{1}{3} - \dots \right) \\
&=   \tan^{-1}(1) - \frac{1}{2} \, \log\left(1+1\right)= \frac{\pi}{4} - \frac{1}{2} \, \log(2)
\end{align}
