# Using Power Series to find this sum

• April 7th 2010, 01:31 PM
paupsers
Using Power Series to find this sum
Can anyone help me on this? I'm trying to find the sum of

$\sum_1^\infty \frac{k}{3^{2k}}$ using power series.

This looks like a geometric sequence to me, but I can't get it in a form that lets me use a power series... any help?
• April 7th 2010, 01:47 PM
Plato
$\sum\limits_{k = 1}^\infty {x^k } = \frac{x}{{1 - x}}$
Differenate to get $\sum\limits_{k = 1}^\infty {kx^{k-1} } = \frac{1}{{(1 - x)^2}}$
Multiply by x $\sum\limits_{k = 1}^\infty {kx^{k} } = \frac{x}{{(1 - x)^2}}$.
In your problem let $x=\frac{1}{9}$.