30.3. Let $|x|<1$. Compute the sum of the series $\sum_{n=1}^{\infty} n x^{n-1}$.
30.3. Answer: $\frac{1}{(1-x)^{2}}$. According to problem 9.6

$$
\sum_{n=1}^{m+1} n x^{n-1}=\frac{(m+1) x^{m+2}-(m+2) x^{m+1}+1}{(1-x)^{2}}
$$

It is also clear that $\lim _{m \rightarrow \infty} x^{m}=0$.