Finding the limit of a sequence

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To determine the limit of the sequence (1+1/n^2)^(n^2), it is established that as n approaches infinity, (1+1/n)^n approaches e. By substituting N=n^2, it follows that (1+1/N)^N also approaches e, suggesting that (1+1/n^2)^(n^2) converges to e as well. The discussion emphasizes that while this reasoning is mathematically sound, it is important to explicitly state the limit notation. It is confirmed that the limits of N and n^2 are the same as n approaches infinity. Ultimately, the limit of the sequence is concluded to be e.
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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.
 
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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.
 
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Yes, the limit of N is the same as the limit of n^2 as n goes to infinity. And what is that limit? It's pretty obvious but you should say it.
 
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I think it would go to e.
Thanks for your help!
 
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Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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