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Homework Help: Advanced Calculus Sequences

  1. Sep 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Determine whether the given limit exists and find their values. Give clear explanations using limit properties.


    2. Relevant equations

    lim n--->∞ (n^2)/n!

    3. 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!
     
  2. jcsd
  3. Sep 19, 2012 #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.
     
  4. Sep 19, 2012 #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?
     
  5. Sep 19, 2012 #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?
     
  6. Sep 19, 2012 #5
    This might be overkill for this problem, but try the squeeze theorem for a rigorous proof.
     
  7. Sep 20, 2012 #6

    jbunniii

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    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?
     
  8. Sep 20, 2012 #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?
     
  9. Sep 20, 2012 #8

    jbunniii

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    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.
     
  10. Sep 20, 2012 #9
    Yes, I am able to use the product theorem. Thanks for the help!
     
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