Show that this sequence converges to 0

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In summary, the conversation discusses a proof that a given sequence converges to 0 and the validity of the proof provided. The speaker suggests another approach using the inequality a*(a+1)*(a+2)*...*(a + n - 1) > a^n.
  • #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
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #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.
 

1. What does it mean for a sequence to converge to 0?

When a sequence converges to 0, it means that as the sequence progresses, the terms get closer and closer to the value of 0. In other words, no matter how small of a number you choose, there will be a point in the sequence where all subsequent terms are less than that number.

2. How do you prove that a sequence converges to 0?

To prove that a sequence converges to 0, you must show that for any given small number, there exists a point in the sequence where all subsequent terms are less than that number. This can be done using the definition of convergence and mathematical techniques such as the epsilon-delta proof.

3. Why is it important to show that a sequence converges to 0?

Showing that a sequence converges to 0 is important because it indicates that the terms in the sequence are approaching a specific value. This can help in understanding the behavior and limits of the sequence, and can also be useful in solving mathematical problems and proving theorems.

4. Can a sequence converge to 0 without actually reaching the value of 0?

Yes, a sequence can converge to 0 without actually reaching the value of 0. This is known as a limit point or an accumulation point, where the terms in the sequence get infinitely close to 0, but never actually reach it.

5. Are there any other ways to show that a sequence converges to 0 besides the epsilon-delta proof?

Yes, there are other ways to show that a sequence converges to 0. Some common methods include using the squeeze theorem, the Cauchy criterion, or the monotone convergence theorem. Each of these methods may be more suitable for different types of sequences, so it is important to choose the most appropriate one for the given sequence.

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