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Advanced Calculus Sequences

  • #1

Homework Statement


Determine whether the given limit exists and find their values. Give clear explanations using limit properties.


Homework Equations



lim n--->∞ (n^2)/n!

The Attempt at a Solution



I know that the limit is 0, but I don't know how to show it in detailed steps. Please help!
 

Answers and Replies

  • #2
jbunniii
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Well, you can immediately cancel an [itex]n[/itex] from the numerator and the denominator. Then try pairing the remaining [itex]n[/itex] in the numerator with one of the factors in the denominator, and see what you can conclude.
 
  • #3
Dick
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There are a lot of ways. Try one. Then someone can help. You have to TRY something. What are some ways you can show a sequence converges?
 
  • #4
I have already canceled the factor of n, and I am stuck at the next step. I have n/(n-1)(n-2)!

Any suggestions what to do next? How do I proof that =0?
 
  • #5
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This might be overkill for this problem, but try the squeeze theorem for a rigorous proof.
 
  • #6
jbunniii
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I have already canceled the factor of n, and I am stuck at the next step. I have n/(n-1)(n-2)!

Any suggestions what to do next? How do I proof that =0?
OK, so you have this:

[tex]\left(\frac{n}{n-1}\right)\left(\frac{1}{(n-2)!}\right)[/tex]

Can you compute the limits of the two factors in parentheses?
 
  • #7
The limit of the first is 1, and the limit of the next is 0? Correct? Then can I simply multiply 1(0)=0. Will that be enough explanation?
 
  • #8
jbunniii
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The limit of the first is 1, and the limit of the next is 0? Correct? Then can I simply multiply 1(0)=0. Will that be enough explanation?
Yes, as long as you have the theorem that the limit of a product is the product of the limits. If not, you will either have to prove that, or find your limit a different way.
 
  • #9
Yes, I am able to use the product theorem. Thanks for the help!
 

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