Limit of a sequence $(1+1/n)^{(n+1)}$

Question

I have to find the limit of a sequence $(1+1/n)^{(1+n)}$. Any help is most welcome.

Answer 1

This is not a complete answer, but a big hint: do you agree that $$ \left( 1 + \frac{1}{n} \right)^{n+1} = \left( 1 + \frac{1}{n} \right)^n \left( 1 + \frac{1}{n} \right) \ ? $$ As $n \to +\infty$, can you read a very famous limit on the right-hand side? The other term won't do you any harm, since it converges to ...

Answer 2

Use $$ \left( 1 + \frac{1}{n} \right)^{n+1} =\exp\left((n+1)\log(1+\frac{1}n)\right) $$ and look at the limit for the exponent: $$ \lim_{n\to \infty} \left((n+1)\log(1+\frac{1}n)\right)=\lim_{n\to \infty} \frac{\log(1+\frac{1}n)}{\frac{1}{n+1}}=\frac{0}{0} $$ so we can try L'Hospitâl and get $$ \lim_{n\to \infty} \frac{-\frac{1}{n^2}\frac{1}{1+1/n}}{-\frac{1}{(n+1)^2}}=\lim_{n\to \infty} \frac{n+1}{n}=1. $$ Therefore your limit is $e^1$.