Hey all,

I have $\displaystyle \sum_{n=0}^{ \infty } a_{n} k^{-n}$ for k=1,2,3... where $\displaystyle a_{n}$ is a term of some infinite sequence {$\displaystyle a_{n}$}

Some manipulation:

$\displaystyle \sum_{n=0}^{ \infty } a_{n} k^{-n} = \sum_{n=0}^{ \infty } \frac {a_{n}} {k^{n}} = \sum_{n=0}^{ \infty } a_{n} \frac {1} {k^{n}} = \sum_{n=0}^{ \infty } a_{n} (\frac {1}{k})^{n} $

Is there anything I can do with this now?

$\displaystyle = \sum_{n=0}^{ \infty } a_{n} \sum_{n=0}^{\infty} (\frac {1}{k})^{n} $

Is it legal to multiply the sums like this?

The second factor looks like a geometric series and I should be able to get the sum of the terms equal to some constant A (if series converges).

$\displaystyle = A \sum_{n=0}^{ \infty } a_{n} $ (for each k)

Does that seem reasonable or did I pull it out of my behind?