<s>[INST] Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Given the geometric series: 
$1 + x^2 + x^4 + x^6 + x^8 + \cdots$
We can recast it as:
$S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$.
This recasting is possible only because there is an infinite number of terms in $S$.
Exactly how is this mathematically possible?
(Related, but not identical, question: General question on relation between infinite series and complex numbers).
 [/INST] There is a finite version of which the expression you have is the limit.
Suppose $S=1+x^2+x^4+x^6+x^8$, then we can put
$S+x^{10}=1+x^2(1+x^2+x^4+x^6+x^8)=1+x^2S$
And obviously this can be taken as far as you like, so you can replace 10 with 10,000 if you choose. If the absolute value of $x$ is less than 1, this extra term approaches zero as the exponent increases.
There is also a theory of formal power series, which does not depend on notions of convergence.
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