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).

Solutions Collecting From Web of "Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$"

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.