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
No comments:
Post a Comment