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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!

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  • $\begingroup$ Please include the question in the body and not just in the title. $\endgroup$ Sep 7, 2011 at 9:43

1 Answer 1

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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.

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    $\begingroup$ Yes! I got it. I integrated $ \dfrac{1}{1+t^3}$ (result: $ \dfrac{1}{3} \left({\frac{\pi}{\sqrt {3}}+\log 2} \right) $ ) which also equals to {your} $ f(t)= 1-t^3+t^6-\ldots $ (result: L.H.S.) . [Sorry for late response. I was busy in solving other problems. Many thanks to you.] $\endgroup$ Sep 7, 2011 at 11:43
  • $\begingroup$ One important fact is the use of Abel's theorem on this result. $\endgroup$
    – DIEGO R.
    Jan 15, 2018 at 6:06

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