Sum of a nearly classic series

Assuming we know that : $$\sum_{n=1}^{+\infty}{\frac{1}{n^2}} = \frac{\pi^2}{6}$$

How do you find the sum of a series where all terms are in this one ?

For instance, how do you prove that ?$$\sum_{n=1}^{+\infty}{\frac{1}{(2n-1)^2}} = \frac{\pi^2}{8}$$

• The question has no answer, except that the sums one can realize are some (but not all) numbers in $[0,\pi^2/6]$. The specific instance in the second part of the question is answered below.
– Did
Oct 14, 2012 at 9:17
• The first sum is absolutely convergent, so you may reorder the terms. Consider splitting into even and odd terms and the second sum follows almost instantly. Oct 14, 2012 at 9:30
• By the way, $\pi^2/8$ in the RHS of the second identity should be replaced by $(\pi^2/8)-1$.
– Did
Oct 14, 2012 at 9:57
• I changed the LHS. Oct 14, 2012 at 11:54

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \sum_{n=1}^{\infty}\frac{1}{(2n)^2} + \sum_{n=1}^{\infty}\frac{1}{(2n-1)^2} = \frac{\pi^2}{6}$$
$$\sum_{n=1}^{\infty} \frac{1}{(2n)^2} = \frac{1}{4}\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{24}$$