Finding power series representations?

Quote:

Originally Posted by sj9110

1/[(1+2x)^2]

Hint: Differentiate, find a power series for the derivative, then integrate the power series

This hint really doesn't make sense to me. Why would you differentiate it first? If anything I feel like you should integrate it first, in which case you would get -1/(2+4x) and you can find a power representation for this. Any advice?

I agree, the hint doesn't make sense. If you start with the power series

$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots$

diffferentiate

$\frac{-1}{(1+x)^2} = - 1 + 2x - 3x^2 + \cdots$

then negate

$\frac{1}{(1+x)^2} = 1 - 2x + 3x^2 + \cdots$

then replace $x$ with $2x$ you'll get the desired power series.