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Given the geometric series:

$1 + x^2 + x^4 + x^6 + x^8 + \cdots$

We can recast it as:

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$S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$.

This recasting is possible ** only** because there is an

Exactly how is this mathematically possible?

(Related, but not identical, question: General question on relation between infinite series and complex numbers).

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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.

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 $S=1+x^2S$.

$S = 1 + x^2 \, S$ is true even in the ring of formal power series. No convergence is needed here.

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