# Sum series

## Homework Statement

[PLAIN]http://img651.imageshack.us/img651/2219/unledybi.jpg [Broken]
can someone teach me how to do it ?
thanks

## The Attempt at a Solution

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$$\sum_{n=1}^∞(3-27k)k^{2n}=(3-27k)\sum_{n=1}^∞(k^2)^n=(3-27k)(\sum_{n=0}^∞(k^2)^n-1)=(3-27k)(\frac{1}{1-k^2}-1)$$
assuming $|k|<1$
$$(3-27k)(\frac{1}{1-k^2}-1)=-7k$$
Now multiply with $(1-k^2)$ and use your favorite CAS program to get
$$-3(-1+9k)k=7(-1+k^2)$$