Let $e=2.17...$ be defined as $$ e = \lim_{n \to \infty} (1+1/n)^n$$ .Prove that $$e= \frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots$$ using induction
hint: $(a+b)^n$ can be expanded using binomial thm and use $${n \choose{k}}+ {n \choose{ k-1}} = {n+1 \choose k}$$
I can show this using taylor expansion of e^x, but my professor wants me to use the hint. I expanded (1+1/n)^n using binomial thm, but have no idea what is the next step. I can't even use the hint :( I see there are duplicate questions but my professor wants us to use the hint! Any ideas? any suggestion or hint is welcomed! Thanks
