The function f(x)=2/(1+9x^2) is represented as a power series f(x)= ∑n=0 to ∞ CnX^n

Question

The function f(x)=2/(1+9x^2) is represented as a power series f(x)= ∑n=0 to ∞ CnX^n

Find the first few coefficients in the power series

C0=

C1=

C2=

C3=

C4=

Find the radius of convergence R of the power series

R=

Answer 1

If a = 2 and r =-9x^2, we have a geometric series which sums to a/(1-r) = 2/(1 -(-9x^2) ) =

2/(1 - 9x^2)

The geometric series is ∑_n=0^∞ 2(-9x²)ⁿ

C0 = 2

C1 =0

C2 = 2 * -9 = -18

C3 = 0

C4 = 2 * (-9)² = 162

As the ratio is -9x^2, |-9x^2| < 1. or 9x^2 < 1, or |x| < 1/3