How do you find a power series representation for f(x)= x/(9+x^2) and what is the radius of convergence?

1 Answer
Jun 27, 2018

x/(9+x^2) = sum_(k=0)^oo (-1)^k x^(2k+1)/3^(2k+2)

with radius of convergence R=3.

Explanation:

Note that:

x/(9+x^2) = x/9 1/(1+(x/3)^2)

Consider the sum of the geometric series:

sum_(k=0)^oo q^k = 1/(1-q)

converging for abs q < 1

Let q=-(x/3)^2 then:

1/(1+(x/3)^2) = sum_(k=0)^oo (-(x/3)^2)^k = sum_(k=0)^oo (-1)^k x^(2k)/3^(2k)

converging for (x/3)^2 < 1, that is for x in (-3,3).

Now:

x/(9+x^2) = x/9sum_(k=0)^oo (-1)^k x^(2k)/3^(2k) = sum_(k=0)^oo (-1)^k x^(2k+1)/3^(2k+2)

with radius of convergence R=3.