I need help with finding a power series representation for the function and determining the radius of convergence.

Question

f(x)=(x/(2-x))³

I'm not sure what the first step would be. Would I first just consider the function as 1/(2-x) and find series notation?

Answer 1

f(x) = x³(1/(2-x)³)

We need to concentrate on converting 1/(2-x) to 1/(2-x)³.

1/(2-x) = 1/(2(1-x/2))

It turns out if you take the derivative twice of 1/(2-x), you get 2/(2-x)³. We only need half of that.

So if you let

h(x) = (1/4) d²/dx²(1/(1-x/2)) = (1/4)d²/dx²∑^inf_n=0 (x/2)ⁿ

Taking the double derivative and then multiplying by x³, you get

x³h(x) = (1/4)∑^inf_n=0(1/2)ⁿn(n-1)xⁿ⁺¹

The radius of convergence:

R= |r| < 1 = |x/2|<1, R=2