How to compute the sum of a series

Question

How to compute the sum of the following series. I have no idea...

The series is $$\sum_{n=1}^\infty(1+1/2+\cdots+1/n)x^n.$$

Answer 1

We have $$ \begin{align} S(x) &= \sum_{n=1}^{\infty} \sum_{k=1}^{n} \frac{x^n}{k} \\ &= \sum_{k=1}^{\infty} \frac{1}{k} \sum_{n=k}^{\infty} x^n \\ &= \sum_{k=1}^{\infty} \frac{1}{k} \frac{x^{k}}{1-x} \\ &= -\frac{\log{(1-x)}}{1-x} \end{align} $$ In general, we have $$ \sum_{n=b}^{\infty} \left( \sum_{k=1}^n a(k) \right) x^n = \frac{1}{1-x} \sum_{k=b}^{\infty} a(k)x^k. $$

Answer 2

Outline: Note that the Maclaurin series for $-\ln(1-x)$ is $$x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}+\cdots,$$ and the Maclaurin series for $\frac{1}{1-x}$ is $$1+x+x^2+x^3+x^4+\cdots.$$ Multiply.