# Math Help - power series

1. ## power series

find a power series representation

f(x)=(x^2) \ (1-2x)^2

how do i do this??

2. Originally Posted by mathlovet
find a power series representation

f(x)=(x^2) \ (1-2x)^2

how do i do this??
Remember that, $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ for $|x| < 1$.

Thus, $\left( \sum_{n=0}^{\infty} x^n \right)' = \left( \frac{1}{1-x} \right)' \implies \sum_{n=1}^{\infty}nx^{n-1} = \frac{1}{(1-x)^2}$ for $|x| < 1$

This gives us,
$\sum_{n=1}^{\infty} nx^{n+1} = \frac{x^2}{(1-x)^2}, |x|<1$

Thus,
$\sum_{n=1}^{\infty} n(2x)^{n+1} = \frac{(2x)^2}{(1-2x)^2} \implies \sum_{n=1}^{\infty} n 2^{n-1} x^{n+1} = \frac{x^2}{(1-2x)^2}, ~ |x| < \tfrac{1}{2}$