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Show that this sequence converges to 0

  1. Apr 22, 2009 #1
    Hello

    I need to show that this sequence converges to 0:
    http://img238.imageshack.us/img238/139/77726816.gif [Broken]

    Here is my work:
    http://img238.imageshack.us/img238/2570/eqlatexlimnrightarrowin.gif [Broken]
    There is n factors in the numerator and in the denominator. We can pair each factor one with another:
    http://img142.imageshack.us/img142/2570/eqlatexlimnrightarrowin.gif [Broken]

    As n goes to infinity the first terms of this limit go to 0, and thus the limit is = 0.

    Is my proof acceptable? Can someone show me another way to do it?
    Thank you
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 22, 2009 #2

    mathman

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    Your proof is faulty since there is no requirement that r be an integer. However you seem to have the general idea in that once n>r, the additional multipliers as n gets bigger are <1 and approach 0.
     
  4. Apr 22, 2009 #3
    thank you for your answer
    can you suggest me another way to do it?
     
  5. Apr 23, 2009 #4

    mathman

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    Not offhand. The approach described seems to be the simplest.
     
  6. Apr 23, 2009 #5
    Notice that a*(a+1)*(a+2)*...*(a + n - 1) > a^n. So 1 / a*(a+1)*(a+2)*...*(a + n - 1) < 1/a^n. See where you can make use of that.
     
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