<s>[INST] Finding the expression of a power series. So I have the infinite sum:
$$\sum_{n=1}^\infty (-1)^{n-1} nx^{2n}$$
I have found in a previous part that the convergence interval is $(-1,1)$.
I know that the sum should equal
$$\frac {x^2}{{(x^2 +1)}^2}$$
when $|x|\lt1$
But I'm not sure how I can show this from the sum.
 [/INST] \begin{align}
\sum_{n=1}^\infty (-1)^{n-1} nx^{2n} = x^2 - {}2x^4 + 3x^6 - 4x^8 + 5x^{10} - \cdots
\end{align}
$$
\begin{array}{cccccccccc}
 n=1 & & n=2 & & n=3 & & n=4 & & n=5 & & \cdots \\[10pt] \hline
x^2 & + & (-x^4) & + & x^6 & + & (-x^8) & + & x^{10} & + & \cdots \\[10pt]
& & (-x^4) & + & x^6 & + & (-x^8) & + & x^{10} & +  & \cdots \\[10pt]
& & & & x^6 & + & (-x^8) & + & x^{10} & + & \cdots \\[10pt]
& & & & & & (-x^8) & + & x^{10} & + & \cdots \\[10pt]
& & & & & & & & x^{10} & + & \cdots \\[10pt]
& & & & & & & & & & \cdots
\end{array}
$$
First find the sum of each row separately. That's easy because each one is a geometric series.
Then find the sum of all those resulting sums. That also easy because that is also a geometric series.
</s>