Thank you to Tom Cassilly and Jacob Sabbagh. Your videos helped straighten out some of the problems I was having with the integral.
Tom's Video on the original Basel's Problem: https://www.youtube.com/watch?v=_clCcNspthI
Jacob's Video on the alternating Basel's Problem: https://www.youtube.com/watch?v=sAve0qtc_N4
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Complex Analysis - Final Project - Alternating Basel's Problem [but with z^4] | NatokHD