Sum series

1. Oct 9, 2011

Suy

1. The problem statement, all variables and given/known data

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

3. The attempt at a solution

Last edited by a moderator: May 5, 2017
2. Oct 13, 2011

susskind_leon

$$\sum_{n=1}^∞(3-27k)k^{2n}=(3-27k)\sum_{n=1}^∞(k^2)^n=(3-27k)(\sum_{n=0}^∞(k^2)^n-1)=(3-27k)(\frac{1}{1-k^2}-1)$$
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
$$-3(-1+9k)k=7(-1+k^2)$$
Now solve for k