Product of power series

Question

Please how to write this series $$\sum_{n=1}^{\infty} \left(1+\frac{1}{2}+\ldots+\frac{1}{n}\right) z^n,\ z\in\Bbb C$$

into a product of power series?

Thank you

Answer 1

Write $$\sum_{n=1}^{+\infty} \left( \sum_{k=1}^n \frac{1}{k}\right)z^n = \sum_{n=1}^{+\infty} \sum_{k=1}^n \frac{z^k}{k} z^{n-k} = \left(\sum_{n=1}^{+\infty} \frac{z^n}{n}\right)\left(\sum_{n=1}^{+\infty} z^n\right)$$