- #1

Vali

- 48

- 0

$$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$

$$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$

I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty } \sqrt[n]{\frac{n!}{n^{n}}}$ which I know how to solve.

So, how to approach the sum such that the result to be $(n! - 1)$ ?