1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove the following series

  1. Jun 5, 2014 #1
    1. The problem statement, all variables and given/known data

    Prove that, by putting x=0 x=∏ in [itex]x^{2}=\frac{\pi^{2}}{3}+4 \sum\limits_{n=1}^\infty \frac{1}{n^{2}} cos(nx) cos(n \pi)[/itex], that [itex]\frac{\pi^{2}}{8}= \sum\limits_{n=1}^\infty \frac{1}{(2n+1)^{2}}[/itex]

    3. The attempt at a solution

    1111.png

    This a solved problem, I've understood the first two parts, and how the even elements of the series were eliminated, but what about [itex]\frac{\pi^{2}}{6} [/itex] and [itex]\frac{\pi^{2}}{8} [/itex]?
     
    Last edited: Jun 5, 2014
  2. jcsd
  3. Jun 5, 2014 #2
    The answer is actually incorrect. The correct answer is ##\frac{\pi^2}{8}-1##.

    Rewrite series 2 in the following way:
    $$\frac{\pi^2}{3}=4\left(\left(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots \right)-\left(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\cdots \right)\right)$$
    $$\Rightarrow \frac{\pi^2}{3}=4\left(1+\sum_{n=1}^{\infty} \frac{1}{(2n+1)^2} -\frac{1}{2^2}\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots \right)\right)\,\,\,\,\,\,(*)$$
    From (3), you have ##\frac{\pi^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots##, put this in ##(*)## and you should be able to reach the correct answer.
     
  4. Jun 5, 2014 #3
    You can turn the equation (2) into an expression for ##\frac{\pi^2}{12}## easily enough, and then ##\frac{\pi^2}{12}+\frac{\pi^2}{6}=\dots##

    (and I disagree with Pranav-Arora - ##\frac{\pi^2}{8}## is correct)

    Edit - Ah I see now what Pranav-Arora meant - the final summation should run from n=0, otherwise, indeed, the LHS should be ##\frac{\pi^2}{8}-1##
     
    Last edited: Jun 5, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Prove the following series
  1. Prove the following (Replies: 10)

Loading...