5
$\begingroup$

I am asked to prove that $$\sum_{n=0}^\infty \frac{1}{(2n+1)^2}=\frac{\pi^2}{8}.$$ However, I am asked to prove it using the fact that $$\frac{\pi}{2}\tan\left(\frac{\pi}{2}z\right)=\sum_{m \text{ odd}}\left(\frac{1}{m-z}-\frac{1}{m+z}\right),$$ where $z\in \mathbb{C}$, which is something I proved in a previous exercise.

My first thought was using the fact that $$\frac{1}{m-z}-\frac{1}{m-z}=\frac{2z}{m^2-z^2}$$ and therefore $$\sum_{m \text{ odd}}\left(\frac{1}{m-z}-\frac{1}{m+z}\right)=\sum_{m \text{ odd}}\frac{2z}{m^2-z^2}=\sum_{n=0}^\infty \frac{2z}{(2n+1)^2-z^2}.$$ This last series is similar to the one I am aiming at, but I don't know how to transform it into the one that I want. Can someone help me?

$\endgroup$
3
  • 7
    $\begingroup$ well perhaps that dividing by $z$ and considering the limit as $z\to 0$... $\endgroup$ May 21, 2022 at 17:48
  • 2
    $\begingroup$ Some of the minus signs in the denominators should be plus signs. $\endgroup$
    – Martin R
    May 21, 2022 at 17:49
  • $\begingroup$ @MartinR thanks $\endgroup$
    – kubo
    May 21, 2022 at 17:53

3 Answers 3

2
$\begingroup$

Choose $z=\frac{a}{2}$ and take the limit as $a\rightarrow 0$.

We know that

$$ \lim_{x\rightarrow 0}\frac{\tan(x)}{x}=1, $$ so $$ \lim_{a\rightarrow 0}\frac{\frac{\pi}{2}\tan(\frac{\pi a}{4})}{a}=\frac{\pi^2}{8}, $$

which could be verified by L'Hospital's rule. We can replace $\tan(\frac{\pi a}{4})$ with the above summation, i.e.,

$$ \frac{\pi}{2}\tan\left(\frac{\pi a}{4}\right)=\sum_{n=0}\frac{a}{(2n+1)^2-\frac{a^2}{4}}.$$

Now, we can substitute and clean up. $$ \begin{align} \lim_{a\rightarrow 0}\frac{\sum_{n=0}\frac{a}{(2n+1)^2-\frac{a^2}{4}}}{a}&=\frac{\pi^2}{8}\\ \lim_{a\rightarrow 0}\sum_{n=0}\frac{1}{(2n+1)^2-\frac{a^2}{4}}&=\frac{\pi^2}{8}\\ \sum_{n=0}\lim_{a\rightarrow 0}\frac{1}{(2n+1)^2-\frac{a^2}{4}}&=\frac{\pi^2}{8}\\ \sum_{n=0}\frac{1}{(2n+1)^2}&=\frac{\pi^2}{8} \end{align} $$

Disclaimer: Thanks to Dan Velleman for his/her insight.

$\endgroup$
9
  • 1
    $\begingroup$ It looks good. Perhaps you need to add an explanation of why you can move the limit inside the sum. $\endgroup$
    – GEdgar
    May 21, 2022 at 20:51
  • $\begingroup$ @GEdgar Thanks! But I don't think I have a very good explanation for this, other than $(2n+1)^2\geq a$ for all $n$? $\endgroup$
    – ck1987pd
    May 22, 2022 at 0:47
  • 2
    $\begingroup$ Very nice approach This is nitpicking, but the beginning limit equation should have a limit on the RHS, as well; it looked odd at first seeing a limit in x being equal to x. $\endgroup$ May 22, 2022 at 8:48
  • $\begingroup$ @AndrewSotomayor I'll fix it! Thanks :) $\endgroup$
    – ck1987pd
    May 22, 2022 at 12:28
  • 1
    $\begingroup$ Yes, it's good now. It would still be nice to justify swapping the limit and sum. You could appeal to the dominated convergence theorem. $\endgroup$ May 22, 2022 at 19:25
2
$\begingroup$

Following the comment of @RaymondManzoni, we obtain continuing OP's approach \begin{align*} \frac{\pi}{4z}\tan\left(\frac{\pi}{2}z\right)=\sum_{n=0}^\infty\frac{1}{(2n+1)^2-z^2}\tag{1} \end{align*}

We can evaluate the right-hand side of (1) at $z=0$. Since $\tan\left(\frac{\pi}{2}z\right)$ has a series expansion for $|z|<1$ we obtain by taking the limit as $z\to 0$ and the tangent series expansion: \begin{align*} \color{blue}{\sum_{n=0}^\infty\frac{1}{(2n+1)^2}}& =\lim_{z\to 0}\frac{\pi}{4z}\tan\left(\frac{\pi}{2}z\right)\\ &=\lim_{z\to 0}\frac{\pi}{4z}\left(\frac{\pi}{2}z+O(z^3)\right)\\ &\,\,\color{blue}{=\frac{\pi^2}{8}} \end{align*}

$\endgroup$
1
$\begingroup$

Definitely not as good as C.Koca's proof because this will just feel like it came out of nowhere (and it's also not the proof you asked for! Just adding it there because might as well, if someone's looking for it). But here's an alternative proof using Fourier series:

Start from the function $f : \mathbb{R} \to \mathbb{R}$ which is $x \mapsto 1 + x/\pi$ on $(-\pi, 0)$, $x \mapsto 1 - x/\pi$ on $(0, \pi)$, and $0$ everywhere else. It's not even continuous but it doesn't matter, you can still compute its Fourier decomposition in the usual way, and get:

$f(x) = \dfrac{1}{2} + \dfrac{4}{\pi^2}\displaystyle\sum_{n \geq 0} \dfrac{\cos\big((2n + 1)x\big)}{(2n+1)^2}$

You can then apply your function at an appropriate value of $x$ to make the right-hand side look like the sum you want it to. And the left-hand side if explicitly given by the original expression for $f$, so you get an equality from which you can deduce the value of your sum :).

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .