MHB Power Series for f(x) and Radius of Convergence

[joshuapeterson]

f(x) = 4x/(x-3)^2
Find the first five non-zero terms of power series representation centered at x = 0.
Also find the radius of convergence.

 

[skeeter]

let $y' = \dfrac{1}{(x-3)^2} \implies y = -\dfrac{1}{x-3} = \dfrac{\frac{1}{3}}{1-\dfrac{x}{3}} = \dfrac{1}{3}\bigg[1+\dfrac{x}{3} + \dfrac{x^2}{3^2} + \dfrac{x^3}{3^3} + \, ... \bigg]$

$y'= \dfrac{1}{3}\bigg[\dfrac{1}{3} + \dfrac{2x}{3^2} + \dfrac{3x^2}{3^3} + \dfrac{4x^3}{3^4}+ \, ... \bigg]$

$f(x) = \dfrac{4x}{(x-3)^2} = \dfrac{4x}{3}\bigg[\dfrac{1}{3} + \dfrac{2x}{3^2} + \dfrac{3x^2}{3^3} + \dfrac{4x^3}{3^4}+ \, ... \bigg] = \dfrac{4x}{3^2} + \dfrac{8x^2}{3^3} + \dfrac{12x^3}{3^4} + \dfrac{16x^4}{3^5}+ \, ... $

I'll leave the 5th non-zero term for you to figure out ...