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).

The $n$th partial sum of your series is
$$
\begin{align*}
S_n   &=   1+x^2+x^4+\cdots +x^{2n}= 1+x^2(1+x^2+x^4+\cdots +x^{2n-2})\\
      &=   1+x^2S_{n-1}
\end{align*}
$$
Assuming your series converges you get that 
$$
\lim_{n\to\infty}S_n=\lim_{n\to\infty}S_{n-1}=S.
$$
Thus $S=1+x^2S$.
