Can anyone help me on this? I'm trying to find the sum of

$\displaystyle \sum_1^\infty \frac{k}{3^{2k}}$ 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?

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- Apr 7th 2010, 01:31 PMpaupsersUsing Power Series to find this sum
Can anyone help me on this? I'm trying to find the sum of

$\displaystyle \sum_1^\infty \frac{k}{3^{2k}}$ 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? - Apr 7th 2010, 01:47 PMPlato
$\displaystyle \sum\limits_{k = 1}^\infty {x^k } = \frac{x}{{1 - x}}$

Differenate to get $\displaystyle \sum\limits_{k = 1}^\infty {kx^{k-1} } = \frac{1}{{(1 - x)^2}}$

Multiply by x $\displaystyle \sum\limits_{k = 1}^\infty {kx^{k} } = \frac{x}{{(1 - x)^2}}$.

In your problem let $\displaystyle x=\frac{1}{9}$.