Proof of the inequality $1+1!+\frac{1}{2!}+\cdots+\frac{1}{n!} >\left( 1+\frac{1}{n}\right)^n$

Question

I would appreciate if somebody could help me with the following problem:

Q: Proof

$$1+1!+\frac{1}{2!}+\cdots+\frac{1}{n!} >\left( 1+\frac{1}{n}\right)^n (n\geq2, n\in \mathbb{N})$$

Answer 1

The right hand side is $$\begin{align} \sum_{k=0}^n{n\choose k}\frac1{n^k}&=\sum_{k=0}^n\frac{n!}{(n-k)!k!}\cdot\frac1{n^k}\\&=\sum_{k=0}^n\frac{n(n-1)\cdots(n-k+1)}{k!}\cdot\frac1{n^k}\\&\le\sum_{k=0}^n\frac{n^k}{k!}\cdot\frac1{n^k}=\sum_{k=0}^n\frac{1}{k!}\end{align}$$ and $\le$ can be replaced by $<$ if a summand with $k\ge2$ occurs.