Sum somewhat similar to Basel problem?

[physstudent.4]

Homework Statement

For a problem in quantum, I am finding the probability of a particle initially in the ground state on a circular loop of length L being in the nth state of the string after it is cut (becomes an infinite square well, and we assume the wavefunction is not disturbed during this cutting). I believe I found this probability, and am now trying to show that the sum over all possible states is 1. This is where I am stuck. I have

Homework Equations

$P_{n=odd}=\frac{16}{\pi^2}\frac{n^2}{(n^2-4)^2}$
$P_{n=even}=0$
And maybe
$\sum_{n=1}^{\infty}=\frac{\pi^2}{6}$

The Attempt at a Solution

The solution I have was getting thus far, and I am fairly confident in it. I do not expect a solution, just a point in the right direction. I did attempt to write my probability as a derivative of n, but that led to another dead end fairly quickly.

 

[physstudent.4]

Nevermind, figured it out!