TITLE: Prove that $ \frac{1}{1}-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\ldots= \frac{1}{3} \left( {\frac{\pi}{\sqrt{3}}+ \log 2} \right)$
QUESTION [8 upvotes]: How do I show that
$$ \frac{1}{1}-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\ldots= \frac{1}{3} \left( {\frac{\pi}{\sqrt{3}}+ \log 2} \right)?$$
This problem belongs to Riemann Theory of Definite Integral, and not to any series summation. I recommend an answer which is to the topic i.e., Riemann Theory of D.I..
Thanks!

REPLY [18 votes]: HINT: First argue out the convergence by alternating series test. Then consider $f(t) = 1 - t^3 + t^6 - \cdots$ where $0 \leq t \lt 1$. Integrate $f(t)$ in two different ways to get to the answer.