Evaluate $\sum_{k=1}^{\infty} (2k+1)\,x^{2k}$,$\,$ for $|x| < 1$ Let $f(x)$ be $\sum_{k=1}^{\infty} x^{2k+1}$. This sum equals $$x + x^3 + x^5 + \dots - x= x(1 + x^2 + (x^2)^2 + \dots) - x = \frac{x}{1-x^2} - x = \frac{x^3}{1-x^2}$$
Now, okay, the original sum is the derivative of $f$, but I don't think it is the conclusion the author expected...?

Yes you are allowed to differentiate, as $f(x)$ is differentiable for every $|x|<1$.
Then
$$
f'(x)=\frac{3x^2}{1-x^2}+\frac{2x^4}{(1-x^2)^2}.
$$
