Represent the function 10x/(3+x) as a power series f(x)=∑ n=0 to ∞ CnX^n

Question

Represent the function 10x/(3+x) as a power series f(x)=∑ n=0 to ∞ CnX^n

C0=

C1=

C2=

C3=

C4=

Find the radius of convergence R=

Answer 1

f(x) = 10x/(3+x) = 10/3x(1-(-x/3)) = (10/3)x/(1-(-x/3)) = (10/3)x∑(-1)ⁿ(x/3)ⁿ = 30∑(-1)ⁿ(x/3)ⁿ⁺¹

C_n = 10(-1)ⁿ/3ⁿ⁺¹

C0 = 10/3

C1 = -10/9

C2 = 10/27

C3 = -10/81

C4 = 10/243

R = -x/3