Power Series Representation of a Function

Question

Find a power series representation for the following functions and determine the interval of

convergence and radius of convergence.

f(x) = 6x/(5x2 - 4x - 1)

Answer 1

f(x) = 6x/(5x² - 4x - 1) = 1/(5x + 1) + 1/(x - 1) = 1/(1 + 5x) - 1/(1- x);

1/(1 + 5x) = 1 + (-5x) + (-5x)² + (-5x)³ + ...+ (-5x)ⁿ + ...; - 1/5 < x < 1/5;

1/(1-x) = 1 + x + x² + ... + xⁿ +...; -1 < x < 1,

So, f(x) = - 6x + 24x²- 126x³ + ... + [(-5)ⁿ - 1]xⁿ +... ; -1/5 < x < 1/5

Answer 2

6x/(5x²-4x+1) = -1/(1-x) - 1/(1-(-5x)) by partial fraction expansion

So you have two series to combine

-∑xⁿ+(-5x)ⁿ = -∑(1+(-5)ⁿ)xⁿ

Two radii of convergence: x and -5x

|x| < 1 --> -1 < x < 1

|-5x| < 1 --> -1 < 5x < 1 --> -1/5 < x < 1/5 Pick this one for interval of convergence since it is smaller

Thus also choosing -5x as the radius of convergence