This is a strictly preliminary question. I hope to elicit some discussion/s which will lead to a more acceptable form for the question on this site.
I'm trying to understand how the study of the following infinite series:
$1 + x^2 + x^4 + x^6 + \cdots$ [1]
and complex numbers in general are related.
Specifically, given the relation:
$1 + x^2 + x^4 + x^6 + \cdots$ = $(1 - x^2)^{-1}$ [2],
and
$1 + 2^2 + 2^4 + 2^6 + \cdots$ = -$\frac{1}{3}$ [3]
in what way do complex number concepts come into play when trying to understand the 'intricacies' of equation [3] above?
(I've come to gain some understanding of the equations above; namely:
(1) the relation [2] is only applicable when |x| < 1;
(2) Euler was generally happy to use relations as [3] in his studies.
But, specifically, in trying to study this problem, where and how are complex numbers employed to come to a deeper understanding of the question?)
Request to potential answerers: I have a very minimal background in real analysis, and almost none of complex analysis. If you may be kind enough to answer / respond so I may understand your answer / response that would be greatly appreciated. :)