Find the sum of power series

Question

I have to find the sum of the following sequence

$$\sum_{n=1}^\infty \frac{x^n}{n-1},\quad x\geq0$$ and

How am I supposed to start?

Thanks.

Answer 1

Hint: Multiply by $x$ and take the derivative.

Answer 2

So what @RobertIsrael means is that look at $f(x) = \sum_{n=1}^\infty \frac{x^n}{n+1}$, let's look at $$g(x) = xf(x) =\sum_{n=1}^\infty \frac{x^{n+1}}{n+1} $$ Differentiating, you get $$g'(x) = \sum_{n=1}^\infty x^n = \frac{x}{1-x}$$ Now integrate $$g(x) = xf(x) =\int \frac{x}{1-x} = -x-\ln\left(\left|x-1\right|\right)$$ then divide by $x$ $$f(x) = -1-\frac{1}{x}\ln\left(\left|x-1\right|\right)$$