Challenge Intermediate Math Problem of the Week 10/5/2017

[PF PotW Robot]

Here is this week's intermediate math problem of the week. We have several members who will check solutions, but we also welcome the community in general to step in. We also encourage finding different methods to the solution. If one has been found, see if there is another way. Occasionally there will be prizes for extraordinary or clever methods. Spoiler tags are optional.

Evaluate the sum of the alternating series $$\sum\limits_{n = 1}^\infty \frac{(-1)^{n-1}}{n^4}$$

(PotW thanks to our friends at http://www.mathhelpboards.com/)

 

[Charles Link]

A very interesting problem. Thank you @PotW Tobor :) I basically googled the answer, where a video showed a related problem of how to compute the ratio of the sum of the p-series divided by the alternating p-series.