• December 10th 2007, 08:02 PM
tbyou87
Find the radius of convergence of R of the following power series.

$\sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}$

I don't know what to do with the $x^{2k}$ term.

Thanks
• December 10th 2007, 08:18 PM
Jhevon
Quote:

Originally Posted by tbyou87
Find the radius of convergence of R of the following power series.

$\sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}$

I don't know what to do with the $x^{2k}$ term.

Thanks

did you try the root test?
• December 10th 2007, 08:20 PM
ThePerfectHacker
Quote:

Originally Posted by tbyou87
Find the radius of convergence of R of the following power series.

$\sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}$

I don't know what to do with the $x^{2k}$ term.

Thanks

Use the root test,
$\left| \frac{k2^k}{3^k}x^{2k} \right|^{1/k} = \frac{2}{3}k^{1/k}x^2$ the limit of $k^{1/k}\to 1$ thus:
$|a_kx^{2k}|^{1/k} \to \frac{2}{3}x^2$.
We require that,
$\frac{2}{3}x^2 < 1 \implies x^2 < \frac{3}{2}$.