What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?

Question

How to solve the following limit question? $$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$

Thanks a lot.

Answer 1

Why don't you want to use the fact that: $$\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e$$ So it will be $$\lim_{n \to \infty}\frac{1}{e^n}\bigg(\left(1 + \frac{1}{n}\right)^n\bigg)^n $$

Answer 2

Pass to the logarithm inside the exponential: $$e^{-n}(1+1/n)^{n^2}=\exp(-n+n^2\ln(1+1/n))$$ Since $\ln(1+x)=x-x^2/2+o(x^2)$ at $0$, you get $-n+n^2\ln(1+1/n)=-1/2+o(1)$ so the sequence $e^{-n}(1+1/n)^{n^2}$ converges and its limit is $e^{-1/2}$.