Question

Alguém saberia resolver esse somatório?

Answer 1

$\displaystyle S=\sum\limits_{k=0}^n(k+1)\binom{n}{k}\\ \\ \\ S=\sum\limits_{k=0}^nk\binom{n}{k}+\sum\limits_{k=0}^n\binom{n}{k}\\ \\ \\ S=\sum\limits_{k=0}^nk\binom{n}{k}+2^n\\ \\ \\ S=\sum\limits_{k=0}^n\frac{n!}{(n-k)!}+2^n$

$\displaystyle S=n!\sum\limits_{k=0}^n\frac{1}{k!}+2^n\\ \\ \\ \text{Debes saber que:}\\ \\ \Gamma(n,x)=(n-1)!\;e^{-x}\sum\limits_{k=0}^{n-1}\frac{x^k}{k!}\Longrightarrow \Gamma(n+1,1)=n!\;e^{-1}\sum\limits_{k=0}^{n}\frac{1}{k!}\\ \\ \\ \Large\boxed{S=e\cdot\Gamma(n+1,1)+2^n}$