Saturday, January 27, 2018

calculus - How to show that limntoinftyleft(frac(1+frac1n2)n2eright)n=1?




I need to find the limit:
lim
So I know that the limit is 1.
Using Squeeze theorem
? \leq \left(\frac{(1 + \frac{1}{n^2})^{n^2}}{e}\right)^n \leq \left(\frac{e}{e}\right)^n \rightarrow\ 1
What should be instead ? ? Is it possible to solve in another way?
Unfortunately, I can't use L'Hôpital Rule or Series Expansion in this task.


Answer



Note that:



\left(\frac{(1 + \frac{1}{n^2})^{n^2}}{(1 + \frac{1}{n^2})^{n^2+1}}\right)^n=\left(1 + \frac{1}{n^2}\right)^{-n}\leq \left(\frac{(1 + \frac{1}{n^2})^{n^2}}{e}\right)^n \leq \left(\frac{e}{e}\right)^n=1




Since:



\left(1 + \frac{1}{n^2}\right)^{-n}\to 1



By Squeeze Theorem:



\lim_{n\to \infty } \left(\frac{(1 + \frac{1}{n^2})^{n^2}}{e}\right)^n=1


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