Radius of Convergence of product of power series

Question

Is the following statement true?

If $P(z)$ is a power series over $\mathbb C$, then $ P(z) $and $P(z)^n$ have same radius of convergence for any positive integer n.

Answer 1

No, it's not true. Consider the power series

$$P(z) = \sqrt{1+z} = \sum_{k=0}^\infty \binom{\frac{1}{2}}{k}z^k.$$

The radius of convergence of $P(z)$ is $1$, since $\sqrt{1+z}$ has a branch point in $-1$. But $P(z)^2 = 1+z$ has infinite radius of convergence.

However, the radius of convergence of $P(z)^n$ is always at least as large as the radius of convergence of $P(z)$, since the radius of convergence is the distance from the centre to the nearest singularity of the represented function, and raising to an integer power does not introduce singularities - it can, however, remove singularities.