Radius of convergence of power series $\sum c_n x^{2n}$ and $\sum c_n x^{n^2}$ I've got a start on the question I've written below. I'm hoping for some help to finish it off.

Suppose that the power series $\sum_{n=0}^{\infty}c_n x^n$ has a radius of convergence $R \in (0, \infty)$. Find the radii of convergence of the power series $\sum_{n=0}^{\infty}c_n x^{2n}$ and $\sum_{n=0}^{\infty}c_n x^{n^2}$.

From Hadamard's Theorem I know that the radius of convergence for $\sum_{n=0}^{\infty}c_n x^n$ is $R=\frac{1}{\alpha}$, where
$$\alpha = \limsup_{n \to \infty} |a_n|^{\frac{1}{n}}.$$
Now, applying the Root Test to $\sum_{n=0}^{\infty}c_n x^{2n}$ gives
$$\limsup |a_nx^{2n}|^{\frac{1}{n}}=x^2 \cdot \limsup |a_n|^{\frac{1}{n}}=x^2 \alpha$$
which gives a radius of convergence $R_1 = \frac{1}{\sqrt{\alpha}}$. Now for the second power series. My first thought was to take
$$\limsup |a_nx^{n^2}|^{\frac{1}{n^2}}=|x| \cdot \limsup |a_n|^{\frac{1}{n^2}}$$
but then I'm stuck. I was trying to write the radius of convergence once again in terms of $\alpha$. Any input appreciated and thanks a bunch.

The radius of convergence of the series $\sum\limits_na_nx^n$ is the only $R_a$ in $[0,\infty]$ such that $|a_n|s^n\to0$ for every $s\lt R_a$ and $|a_n|s^n\not\to0$ for every $s\gt R_a$.
If $a_{2n}=c_n$ and $a_{2n+1}=0$, then for every $s\lt\sqrt{R_c}$, $|a_n|s^n\to0$ and for every $s\gt\sqrt{R_c}$, $|a_{2n}|s^{2n}=|c_n|(s^2)^n\not\to0$, thus $R_a=\sqrt{R_c}$.
If $a_{n^2}=c_n$ and $a_k=0$ for every $k$ not a square, then for every $s\lt1$, $|a_{n}|s^{n}\to0$ and for every $s\gt1$, $|a_{n^2}|s^{n^2}=|c_n|s^{n^2}\not\to0$ (this last part uses the fact that $R_c$ is finite), thus $R_a=1$.
