Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$

Question

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$

I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] =r(x+1)^{r-1}=\sum_{n=1}^\infty \binom{r}{n}nx^{n-1}$

However, I am not sure how to "get rid" of the binomial term.

Answer 1

Note that for $|x|<1$, we have $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$

Taking derivatives of both sides

$$\sum_{n=0}^\infty nx^{n-1}=\frac{1}{(1-x)^2}$$

On the LHS, when $n=0$, the term vanishes. Thus

$$\sum_{n=1}^\infty nx^{n-1}=\frac{1}{(1-x)^2}$$