<s>[INST] A binomial inequality with factorial fractions: $\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$ Prove that $$\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$$ for $n>1 , n \in  \mathbb{N}$.
 [/INST] The binomial expansion tells you the following $$(1+\frac{1}{n})^n= 1^n + n*1^{n-1}*\frac{1}{n}+\binom{n}{2}*1^{n-2}*(\frac{1}{n})^2 + \cdots +(\frac{1}{n})^n $$ $$= 1+ 1+ \frac{1}{2!}(1-{1\over n}) + \cdots + \frac{1}{n!}(1-\frac{1}{n})(1-\frac{2}{n})\cdots (1 - \frac{n-1}{n})$$ $$\lt 1+1+\frac{1}{2!}+\cdots+\frac{1}{n!} $$
I recommend you to do the calculation to see how it does this, it really helps in the future.
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