Convert a closed-form generating function to a sequence

Question

I need some help with an assignment question:

I must determine the sequence generated by the following generating function: $2x^3 \over 1 - 5x ^ 2 $

In class we have only gone from the sequence to the closed form so, I am not really sure how to begin on this. I feel like I should begin by separating the functions so that we are working with $ 2x^3 \cdot {1 \over 1-5x^2}$ which is closer to the form that I am used to seeing come from the sequence. We would get these closed-form functions out from the sequence using a table, so logically going the other way should work the same way. In my table, I have nothing of the form $1 \over 1-ax^2$.

Thank you for any assistance.

Answer 1

You're almost there, there's just a little jump you need to make. Remember that, using the formula of a sum of a geometric series, we can state $$ \frac{1}{1-z}=\sum_{n=0}^{\infty}z^n $$ Now, just set $z=5x^2$ to get $$ \frac{2x^3}{1-5x^2}=2x^3\sum_{n=0}^{\infty}\left(5x^2\right)^n $$ I'll let you take it from there...