8
$\begingroup$

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!

  • 0
    Please include the question in the body and not just in the title.2011-09-07

1 Answers 1

18

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.

  • 0
    One important fact is the use of Abel's theorem on this result.2018-01-15