convergent (a nice question)

Question

Find a power series representation for the function g(x) = (x^2) / ( (1 + 3x)^2) about x = 0 where the index of the sum starts from n = 0. Do not forget to write the interval of convergence

Answer 1

Since the expansion is about x = 0, a good idea is to rewrite as

 x^2 / (1 + 3 x) ^2  

This suggests that the expansion of 1 / (1 - x) ^2 is of interest.

But,     1 / (1 -x) ^2 is the derivative of 1 / (1 -x) = 1 + x + x^2 + x^3 + x^4 + …

Differentiating term by term gives:

 1 / (1 -x )^2 = 1 + 2 x + 3 x^2 + 4 x^3  + …

Putting these elements together yields the power series

 x^2 [ 1 – 2 (3x) + 3 (3x)^2 – 4 (3x)^3 + …

The interval of convergence is [-1/3 , 1/3)