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.