A limit about $\left(1+\frac{1}{n}\right)^{n}$?

Question

Here is my question:

$$\displaystyle\lim_{n\rightarrow \infty}n^2\left[\left(1+\frac{1}{1+n}\right)^{n+1}-\left(1+\frac{1}{n}\right)^{n}\right]=?$$

Any hints will be fine. Thank you!

Answer 1

HINT

Note that (to be made more rigorous)

• $(1+\frac{1}{1+n}) ^{n+1}=e^{(n+1)\log\left(1+\frac{1}{1+n}\right)}\sim e^{(n+1)\left(\frac{1}{1+n}-\frac{1}{2(1+n)^2}\right) }=e^{1-\frac{1}{2(1+n)}}\sim e\left(1-\frac{1}{2(1+n)}\right)$

• $(1+\frac{1}{n}) ^{n}=e^{n\log\left(1+\frac{1}{n}\right)}\sim e^{n\left(\frac{1}{n}-\frac{1}{2n^2}\right) }=e^{1-\frac{1}{2n}}\sim e\left(1-\frac{1}{2n}\right)$

then

$$n^2\left[\left(1+\frac{1}{1+n}\right)^{n+1} - \left(1+\frac{1}{n}\right)^{n}\right]\sim e\cdot n^2\left(\frac{1}{2n}-\frac{1}{2(1+n)}\right)=e\cdot n^2\left(\frac{2}{4n^2+4n}\right)$$

Answer 2

Here is "any" hint:

Set $x=1/n$ and you will have $$ \lim_{x\to 0^+}\frac{1}{x^2}\biggl(\Bigl(1+\frac{x}{1+x}\Bigr)^{1+1/x}-(1+x)^{1/x}\biggr). $$ The expression $$ \Bigl(1+\frac{x}{1+x}\Bigr)^{1+1/x}-(1+x)^{1/x} $$ in the right-hand side can be expanded around $x=0$ (perhaps after defining it as $0$ at $x=0$).

Answer 3

Hint: Use $$n^2\left[\left(1+\frac{1}{1+n}\right)^{n+1}-\left(1+\frac{1}{n}\right)^{n}\right]=\frac{\left(1+\frac{1}{1+n}\right)^{n+1}-\left(1+\frac{1}{n}\right)^{n}}{\frac1{n^2}}$$ and then use L'Hopital's Rule.