How do you find a power series representation for f(x)=x9+x2 and what is the radius of convergence?

1 Answer
Jun 27, 2018

x9+x2=k=0(1)kx2k+132k+2

with radius of convergence R=3.

Explanation:

Note that:

x9+x2=x911+(x3)2

Consider the sum of the geometric series:

k=0qk=11q

converging for |q|<1

Let q=(x3)2 then:

11+(x3)2=k=0((x3)2)k=k=0(1)kx2k32k

converging for (x3)2<1, that is for x(3,3).

Now:

x9+x2=x9k=0(1)kx2k32k=k=0(1)kx2k+132k+2

with radius of convergence R=3.