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Homework Help: Proof with inequations.

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data
    I need to prove the following for all natural numbers:

    1/(x+1) < ln(x+1) - ln(x) < 1/x


    2. Relevant equations



    3. The attempt at a solution
    It's in the part of the mean value theorem problems, I try using it didn't go any where. I tried thinking of other ways, but nothing seems to work.
     
  2. jcsd
  3. Dec 2, 2008 #2

    HallsofIvy

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    Science Advisor

    The first thing I would do is write ln(x+1)- ln(x) as ln((x+1)/x). Are you saying that x must be a positive integer? Then I would assume that you should use proof by induction, not the mean value theorem!
     
  4. Dec 2, 2008 #3

    lurflurf

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    Homework Helper

    Induction would work nicely, but taking advantage of your section hint try
    log(x+1)-log(x)=[log(x+1)-log(x)]/[(x+1)-x]=log'(x+t)=1/(x+t)
    where t is some number such that 0<t<1
    so the problem is reduced to showing that
    when ever 0<t<1
    1/(x+1)<1/(x+t)<1/(x+0)
     
  5. Dec 2, 2008 #4
    I got it with the mean value theorem, but because that's for all real numbers, can I say it's true for natural numbers too because it is a subset of the real numbers? Thanks for the help.
     
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