Find the power series representation of f(x)g(x) given f & g as defined below?

Let's look at series for 1/(1−x) and its derivatives:

1/(1−x) = ∑ x^n

1/(1−x)² = ∑ n x^(n−1)

2/(1−x)³ = ∑ (n−1)n x^(n−2)

f(x) = 2 ∑ x^n = 2/(1−x)

g(x) = ∑ n x^n = x/(1−x)²

f(x) g(x) = 2x/(1−x)³ = ∑ (n−1)n x^(n−1) = ∑ n(n+1) x^n

f(x)g(x) = 2∑(r=0,∞) x ͬ * ∑(s=0,∞) sxˢ = 2∑∑(r,s=0,∞) sx ͬ ⁺ˢ

Coefficient of xⁿ term is 2 x sum of all s for which r+s=n i.e 2(0+1+2+…+n) = n(n+1)

∴ f(x)g(x) = ∑(n=0,∞) n(n+1)xⁿ