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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. :)

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    This falls under the purview of series regularization and summability theory. Look at e.g. Abel, Cesàro or Ramanujan summation. The connection to complex analysis comes through analytic continuation of the sort of functional expansions defining any particular summation method.2011-09-30

2 Answers 2