Analytic continuation of $\sum z^n/n$ I have the following function
$$f(z)=\sum_{n=1}^\infty \frac{1}{n}z^n$$
It is easy to see that it converges for $|z|<1$ (root test, for example).
How can it be analytically continuated beyond the unit circle?
The usual trick of relating it to a geometric series doesn't seem to work.
Any idea or hint would be appreaciated!

By integrating the geometric series term by term, you can easily obtain:
$$f(z) = -\log(1-z)$$
However, if you cannot sum a series into a closed form expression, you can sometimes still analytically continue the series to another series with a larger radius of convergence. E.g. if you substitute $z = \frac{w}{2-w}$ and re-expand in powers of $w$, what is then the radius of convergence in the $w$-plane and how does this region map back to the $z$-plane?
