Finding the limit of a sequence

[Caiti]

Homework Statement

How do you determine if the limit of (1+1/n^2)^(n^2) exists and what it is?
This cannot use logarithms at any point.

Homework Equations

(1+1/n)^n --> e

The Attempt at a Solution

Let N=n^2
Given (1+1/N)^N --> e, then (1+1/n^2)^(n^2) must --> e also.
Is this allowed though? Do I need to put restrictions on N?
I was thinking that I might need to show that N and n^2 have the same limit on their own, but since I have created N, it's limit is obviously that of n^2.

 

[Simon Bridge]

Which limit do you mean?

Let N=n^2
Given (1+1/N)^N --> e, then (1+1/n^2)^(n^2) must --> e also.
Is this allowed though? Do I need to put restrictions on N?

... I cannot tell what the person marking you work will or will not allow. It is OK mathematically - except you need the "lim" part of the notation.

 

[HallsofIvy]

Yes, the limit of N is the same as the limit of [itex]n^2[/itex] as n goes to infinity. And what is that limit? It's pretty obvious but you should say it.

 

[Caiti]

I think it would go to e.
Thanks for your help!