How do you find the sum of this infinite series?

Question

The actual series that I want to solve: $$ \sum_{n=2}^{\infty}\frac{n 2^{n-1} + \frac{n(n-1)}{1\cdot 2}2^{n-2}+\frac{n(n-1)(n-2)}{1\cdot 2\cdot 3}2^{n-3}+\cdots+n 2^1+1}{3^n} $$

Answer 1

Hint: Note that $$(2+1)^n = 2^n + n 2^{n-1} + \binom{n}{2} 2^{n-2} + \binom{n}{3} 2^{n-3} + \cdots + n 2^1 + 1.$$

Answer 2

Use binomial theorem to rewrite the numerator as $$\sum_{k=1}^n \binom nk 2^{n-k} = \sum_{k=0}^n \binom nk 2^{n-k} - 2^n = (1+2)^n-2^n=3^n-2^n$$.

Answer 3

HINT

Use Binomial Theorem to write the numerator as $3^n-2^n$ and then simply use the sum of geometric series $$\sum_{n=0}^{\infty}ar^n=\frac{1}{1-r}, \ |r|<1$$