f(x)=xg(x), where g(x)=1/(1-x⁵).
Consider g(y)=1/(1-y).
We can generate a Maclaurin series for g(y):
g⁽⁰⁾(y)=(1-y)⁻¹; g⁽⁰⁾(0)=1
g⁽¹⁾(y)=(1-y)⁻²; g⁽¹⁾(0)=1
g⁽²⁾(y)=2!(1-y)⁻³; g⁽²⁾(0)=2!
...
g⁽ⁿ⁾(y)=n!(1-y)⁻⁽ⁿ⁺¹⁾; g⁽ⁿ⁾(0)=n!1
General Maclaurin term: yⁿ
Series is 1+y+y²+y³+...
Substituting x⁵ for y:
g(x)=1+x⁵+x¹⁰+x¹⁵+...
f(x)=x+x⁶+x¹¹+x¹⁶+...
NOTE: This series requires -1<x<1 to be effective. x/(1-x⁵) → ∞ as x→1 so is not a continuous function of x. The series diverges or oscillates when x>1 or x≤-1. It therefore has a limited application.