Solve $\sum nx^n$
Solution 1:
Convergence of the series below is assumed throughout.
$$\begin{align} \sum\limits_{n\ge0}nx^n&=\sum\limits_{n\ge0}x(x^n)' &\text{integrate } nx^{n-1}\\ &=x\sum\limits_{n\ge0}(x^n)' &\text{factor } x \,\text{out}\\ &=x\left(\sum\limits_{n\ge0}(x^n)\right)' &\text{differentiate the whole series} \\ &=x\left(\frac1{1-x}\right)' &|x|<1\\ &=\frac x{(1-x)^2} &\text{differentiate }\frac {1}{1-x} \end{align}$$
Solution 2:
Hint: The basic idea that we can switch $\frac{d}{dx}$ and $\sum$ in any compact subset of the disc of convergence for the power series.