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Proving a seqeunce converges

  • Thread starter ehrenfest
  • Start date
1,996
1
[SOLVED] proving a seqeunce converges

1. Homework Statement
Prove that the sequence [itex]\frac{n!}{n^n}[/itex] converges to 0.


2. Homework Equations



3. The Attempt at a Solution
Given [itex]\epsilon > 0[/itex], how do I find N? We know that the nth term is less than or equal to
[tex]\left(\frac{n-1}{n}\right)^{n-1}[/tex]
but that really does not help.
 

Answers and Replies

tiny-tim
Science Advisor
Homework Helper
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… in a fight, always pick off the smallest ones …

Hi ehrenfest! :smile:

Hint: write it (1/n)(2/n)(3/n)…(n/n);

the early bits go down much faster than the later bits; so can you see a way of splitting off some of the early bits, and prove that they tend to zero? :smile:
 
1,996
1
Hi ehrenfest! :smile:

Hint: write it (1/n)(2/n)(3/n)…(n/n);

the early bits go down much faster than the later bits; so can you see a way of splitting off some of the early bits, and prove that they tend to zero? :smile:
Yes, I was trying to do something like that. If I can show that any factor in that product goes to 0, then the whole thing has to because the rest will be less than 1. If I collect the first n/2 ceiling terms, then I get (1/2)^(n/2) if n is even, and that needs to got to zero, doesn't it? I see, thanks. When n is odd we get something similar.
 
tiny-tim
Science Advisor
Homework Helper
25,789
249
That's right! :smile:

(And don't forget, your proof should begin something like "For any epsilon, choose N such that 2^-N …")
 

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