# Evaluate the series 1+1/9+1/(25)+1/(49)+cdots

Evaluate the series
$1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+\cdots$
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beshrewd6g
One approach is as follows:
Let $f\left(t\right)$ denote a triangle wave with the precise form described here (we'll take L=1). As you can see in the link, f can be expanded into its Fourier series as
$f\left(t\right)=\frac{8}{{\pi }^{2}}\sum _{n=1}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n-1}}{\left(2n-1{\right)}^{2}}\mathrm{sin}\left(\left(2n-1\right)\pi t\right)$
Now, just plug $t=\frac{1}{2}$ into both sides of the equation above.
###### Did you like this example?
Denisse Fitzpatrick
Hint: On way to evaluate your series is using Parseval theorem. At first you need to find the fourier series of a function like $f\left(x\right)=x$ in $\left[-\pi ,\pi \right]$ and then applying Parseval theorem.