radius of convergence of $\sum_{n=0}^{\infty} (2n+1)(2x)^{2n}$

Question

Determine the radius of convergence of the following power series $\sum_{n=0}^{\infty} (2n+1)(2x)^{2n}$.

Is the following correct?

$\sum_{n=0}^{\infty} (2n+1)(2x)^{2n} = \sum_{n=0}^{\infty} (2n+1)2^{2n}(x^2)^n = \sum_{n=0}^{\infty} (2n+1)4^n(x^2)^n$

$\operatorname{lim sup}_{n \to \infty} \sqrt[n]{(2n+1)4^n} = \operatorname{lim sup}_{n \to \infty} \sqrt[n]{(2n+1)} \cdot \operatorname{lim sup}_{n \to \infty} \sqrt[n]{4^n} = 1\cdot 4=4$

Answer 1

Hint: Use a substitution $t=x^2$.