MHB Johnathan's question at Yahoo Answers (Power series representation)

[Fernando Revilla]

Here is the question:

am studying for a Cal 2 final and I am having a lot of trouble with this one example. Find the power series representation for the function and the radius of convergence. I understand the concepts of power series representations and radii of convergence but I am not sure how to go about solving this problem. f(x) = 1/((2 + x)^2)
I've thought about maybe a partial fraction, but that wouldn't work, then I've thought about making this into f(x) = 1/4 * 1/1-(x^2 + 4x) and setting the x^2 and 4x to my a[n] function but I am not sure if this is correct or how to do it.

Here is a link to the question:

Help with this power series representation? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Johnathan,

Using the geometric series: $$g(x)=\dfrac{1}{x+2}=\dfrac{1}{2}\dfrac{1}{1+ \frac{x}{2}}=\dfrac{1}{2}\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{2^n}\;(|x|<2)$$
Using the uniform convergence of the power series on all $[-\rho,\rho]\subset (-2,2)$: $$g'(x)=-\frac{1}{(x+2)^2}=\sum_{n=1}^{\infty}\frac{(-1)^nnx^{n-1}}{2^{n+1}}\;(|x|<2)$$ As a consequence, $$f(x)=\dfrac{1}{(x+2)^2}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}nx^{n-1}}{2^{n+1}}\;(|x|<2)$$