Solution
= Solution
Apply the root test to the terms $a_nz^{n^2}$:
$$
|a_nz^{n^2}|^{1/n}
=|a_n|^{1/n}|z|^n.
$$
This tends to zero when $|z|<1$ and to infinity when $|z|>1$. On $|z|=1$, the series converges absolutely because $\sum|a_n|$ converges: the point $1$ lies strictly inside the original radius two. Thus the lacunary power series has radius
$$
\boxed{1}
$$
and in fact converges on its entire boundary circle.