# # lim_(n->oo)(1+1/n)^n = # ?

It is obvious that:

Using Bernoulli's inequality again:

Using inequality of arithmetic and geometric means:

it follows:

Finally,

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Thus, we undo the natural logarithm to get:

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so

Finally

and

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The limit of ( (1 + \frac{1}{n})^n ) as ( n ) approaches infinity is equal to ( e ), where ( e ) is Euler's number, approximately ( 2.71828 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- For what values of #r# does the sequence #a_n = (nr)^n# converge ?
- How do you use the direct comparison test to determine if #Sigma 1/(n!)# from #[0,oo)# is convergent or divergent?
- How do you test for convergence of #Sigma (3n-7)/(10n+9)# from #n=[0,oo)#?
- How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n))/(sqrt(n+4))# from #[1,oo)#?
- How do you determine if the improper integral converges or diverges #int [(x^3)( e^(-x^4) )] dx# from negative infinity to infinity?

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