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

Is the following correct?

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

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

  • 3
    $\begingroup$ Continue, tell us what the radius of convergence is... (it's not $4$, but ...???) $\endgroup$ – GEdgar Jun 4 '12 at 14:11

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


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