Radius of convergence of power series where power increases by increments of 2 I know that to determine the radius of convergence of the series
$$ \sum_{n=0}^\infty a_nx^n $$
I need to find
$$ \lim_{k\rightarrow \infty} \left| \frac{a_{k+1}}{a_k} \right| = c$$
Then the radius of convergence $R$
$$R = \frac{1}{c}$$
However how do I calculate the radius of a convergence for the series
$$ \sum_{n=0}^\infty a_nx^{2n} $$
Or more generally
$$ \sum_{n=0}^\infty a_nx^{Bn}, \quad B\in\mathbb{N} $$

Consider the series:
$\begin{equation*}
\sum_{n \ge 0} a_n x^{B n}
\end{equation*}$
From the respective theory, you know that for the series:
$\begin{equation*}
\sum_{n \ge 0} a_n y^n
\end{equation*}$
there is a radius of convergence $R$ such that it converges if $\lvert y \rvert < R$ and diverges whenever $\lvert y \rvert > R$. Now you can use the comparison test (pick $y_0$ so it is $\lvert y_0 \rvert < R$ and compare with the original series at $x_0 = y_0^{1/B}$ to prove convergence; pick a larger one to prove divergence similarly) to show that your original series converges if $\lvert x \rvert < R^{1/B}$ and diverges whenever $\lvert x \rvert > R^{1/B}$.
