power expansion

[walter9459]

1. The problem statement, all variables and given/known data
I am trying to do our the review exam our teacher posted to study for a test and I am having difficulty trying to figure out where to start and what to do. Our teacher lost me when he was explaining this section. Please help!



2. Relevant equations
Write the power expansion for a given function.

x
-------------
(1-x)(1-x^2)


3. The attempt at a solution I wasn't sure where to start or what I need to do!

[marcuss]

did u try F.O.I.L?(first outer inner last) i believe that is all it is asking

[walter9459]

Thanks! But that is not what he is looking for. I need to come up with a summation. One example he worked for us in class was e^(-x^2) = summation (-1)^n [(x^(2n))/n!].

[HallsofIvy]

Since this is in the "Calculus and Beyond" section I would rather interpret that as expanding the function in a power series. One way to do that is to find the Taylor's series for the function. Another way is to use the fact that
\sum_{n=0}^\infty r^n= \frac{1}{1- r}
to interpret 1/(1- x) and 1/(1-x2) as geometric series with r= x and r= x2. Multiply those together (be careful with that) and multiply the result by x (easy).

[walter9459]

Sorry to be so dense but I really have hit a wall where this concept is concerned. I understand what you are saying but not sure what you meant to do next. I really need to understand this concept as I have a test coming up! Thanks!

[HallsofIvy]

Have you written 1/(1-x) and 1/(1- x2) as power series as I said? That is the first step.

[walter9459]

Please accept my apologies. I had been studying all day and had hit a wall. I stepped away and when I came back, it all made sense. Your assistance was greatly appreciated! Thank you for all your help!