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anonymous

  • 5 years ago

does this sequence converge or diverge? (n!)/(n^n) and the reason

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  1. sandra
    • 5 years ago
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    this converges to zero since eventually n! expands into n*....n-(n-1) [i.e. 1], while the last multiple of n^n is still n. i.e. once n is big enough the last divisor is obviously bigger enough to bring the number to zero

  2. anonymous
    • 5 years ago
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    hey sandra you dont have a better reason?

  3. anonymous
    • 5 years ago
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    i dont buy this argument

  4. anonymous
    • 5 years ago
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    how about consider x! / x^x

  5. anonymous
    • 5 years ago
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    for your father

  6. anonymous
    • 5 years ago
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    what you wrote isnt obvious , sorry

  7. anonymous
    • 5 years ago
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    we can make x! continuous using gamma function

  8. sandra
    • 5 years ago
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    hmmm, not sure about that =) can you explain?

  9. anonymous
    • 5 years ago
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    well you didnt show it converges

  10. sandra
    • 5 years ago
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    yeah I know, I gave the best reasoning I could heh

  11. anonymous
    • 5 years ago
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    hehe

  12. anonymous
    • 5 years ago
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    Did you guys try the ratio test?

  13. sandra
    • 5 years ago
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    i for one did not :p

  14. anonymous
    • 5 years ago
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    ratio test is for series, this is a sequence

  15. anonymous
    • 5 years ago
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    or maybe im wrong, i havent see ratio test for sequences

  16. anonymous
    • 5 years ago
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    how about a sandwich theorem

  17. anonymous
    • 5 years ago
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    n*n >= n * n-1 n*n*n > = n * n-1 n - 2

  18. anonymous
    • 5 years ago
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    so by induction we have

  19. anonymous
    • 5 years ago
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    n * n-1 * n-2 / n*n*n < = 1

  20. anonymous
    • 5 years ago
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    err

  21. anonymous
    • 5 years ago
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    it is strictly less than 1

  22. anonymous
    • 5 years ago
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    n* n > (n-1)(n-2) for n >= 1

  23. anonymous
    • 5 years ago
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    ok so divide out the first n

  24. sandra
    • 5 years ago
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    ok, spolier alert... this is a bit beyond my current understanding, but I think the by induction track sounded good... http://www.physicsforums.com/showthread.php?t=195508

  25. sandra
    • 5 years ago
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    the root solution is way over my head

  26. anonymous
    • 5 years ago
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    sandra i made you a hero

  27. sandra
    • 5 years ago
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    omg lol! yay =)

  28. sandra
    • 5 years ago
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    I'm already your fan, so not much I can do :p

  29. anonymous
    • 5 years ago
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    How do you give points to people?

  30. sandra
    • 5 years ago
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    you "become their fan"

  31. sandra
    • 5 years ago
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    I think they should add per answer type ratings

  32. sandra
    • 5 years ago
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    at least in addition

  33. sandra
    • 5 years ago
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    something like "best answer", or "thanks" or something. one more thing to collect :p

  34. anonymous
    • 5 years ago
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    I can't find "become their fan"

  35. anonymous
    • 5 years ago
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    this is a tough question

  36. anonymous
    • 5 years ago
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    see , the series 1/n diverges, but the sequence 1/n converges to zero when you take the limit

  37. sandra
    • 5 years ago
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    hmmm jkwon, should be right by the person's name , e.g. right above my reply HERE

  38. anonymous
    • 5 years ago
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    \[x + x = 2xx\]

  39. anonymous
    • 5 years ago
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    we know n! > n-1 * n-2

  40. anonymous
    • 5 years ago
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    i think we can use the fact that its less than 1 and decreasing

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