Generating Function for $a_n=\frac{1}{n}$

Question

I've been stumped by one of the exercises in the book "Number Theory" by George E. Andrews and I can't seem to figure it out. The exercise asks for the generating function for the sequence $a_n=\frac{1}{n}$, starting at $a_1$, which is defined as $$f(x)=\sum_{n=1}^\infty \frac{x^n}{n}$$ which it tells me will turn out to condense down to $$f(x)=-\ln(1-x)$$ I have no idea how to go from the top equation to the bottom. I already tried multiplying both sides of the top by $1-x$ in the hopes of getting a telescoping sum, but it just gets more complicated. Help?

Answer 1

One takes the geometric series $$\sum_{n=0}^\infty x^n=\frac1{1-x}$$ for $|x|<1$. One then integrates termwise.