function of 3x/(5+x) as power series

Question

Represent the function  3x/(5+x) as power series f(x)= ∑^inf_n=0 c_n xⁿ

Answer 1

3x / (5+x) = 3x / [5(1 - (-x/5))] = (3/5)[ x / (1 - (-x/5)]

x / (1 - (-x/5)) is the sum of the Geometric Series with first term a = x and common ratio r = -x/5.

So, we have: (3/5)∑_{(n=0 to infinity)} arⁿ = (3/5)∑_{(n=0 to infinity)} x(-x/5)ⁿ = 3∑_{(n = 0 to infinity)} [(-1)ⁿxⁿ⁺¹/5ⁿ⁺¹]