Friday, 3 January 2020

calculus - Why is(1+frac1n)n=e when n goes to infinity?




Why is limn(1+1n)n=e?



I think it involves n=01k!=e but not sure how to get from one to the other.


Answer




Have you tried expanding by the binomial theorem? ;-)
(1/n+1)^n=\sum_{k=0}^n\binom{n}k\frac1{n^k}=\sum_{k=0}^n\frac{n!}{k!(n-k)!}\frac1{n^k}then as n\to\infty we find n!/(n^k (n-k)!)\to1 hence we have:\lim_{n\to\infty}\sum_{k=0}^n\frac{n!}{k!(n-k)!\cdot n^k}=\sum_{k=0}^\infty\frac1{k!}\equiv e


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