Existence of a convergent series with a "sub-series" of smaller radius of convergence.

Question

This is a past qualifying exam question:

True or false? There is a sequence of complex numbers $\{a_n\}_{n=0}^\infty$ and strictly increasing sequence of integers $\{p_n\}$ with $p_n \ge n$, such that the radius of convergence of $$\sum_{n=0}^\infty a_nz^n$$ is one, but $$\sum_{n=0}^\infty a_n z^{p_n}$$ is less than one.

I can see that if we took the "sub-series" to be $\sum a_{p_n} z^{p_n}$, then the opposite would be true: that the radius of convergence for the "sub-series" would be $\ge$ the radius of convergence for the regular series. But this is not the problem at hand.

Answer 1

The answer is no. When $|z|<1 $, $|z|^n \geq |z|^{p_n}$, therefore the series $ a_nz^{p_n} $ is dominated by the series $ a_nz^n $ which is absolutely convergent.