Show that this sequence converges to 0

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Discussion Overview

The discussion revolves around demonstrating that a specific sequence converges to 0. Participants explore various methods of proof, including mathematical reasoning and alternative approaches, while addressing the validity of the initial proof presented.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • The original poster presents a proof involving pairing factors in the numerator and denominator, suggesting that as n approaches infinity, the limit approaches 0.
  • Some participants challenge the original proof, noting that it incorrectly assumes r must be an integer, but acknowledge the general idea that additional multipliers become less than 1 and approach 0 as n increases.
  • One participant suggests an alternative approach by comparing the product of terms to a power of a, indicating that the sequence can be bounded above by a simpler expression.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the original proof, with some acknowledging the general concept while others seek alternative methods. No consensus on a definitive proof exists.

Contextual Notes

The discussion highlights potential limitations in the assumptions made about the parameters involved in the sequence, particularly regarding the nature of r.

fishingspree2
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Hello

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

Here is my work:
http://img238.imageshack.us/img238/2570/eqlatexlimnrightarrowin.gif
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

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
 
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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.
 
mathman said:
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.

thank you for your answer
can you suggest me another way to do it?
 
fishingspree2 said:
thank you for your answer
can you suggest me another way to do it?


Not offhand. The approach described seems to be the simplest.
 
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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