How to compute the sum of a series How to compute the sum of the following series.
I have no idea...
The series is $$\sum_{n=1}^\infty(1+1/2+\cdots+1/n)x^n.$$

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
$$ \sum_{n=b}^{\infty} \left( \sum_{k=1}^n a(k) \right) x^n = \frac{1}{1-x} \sum_{k=b}^{\infty} a(k)x^k. $$
