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Homework Help: Trying to prove that for x real, ln(1+x)<=x

  1. Jan 4, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi everyone,
    I'm trying to proove that for x real, ln(1+x)<=x

    2. Relevant equations



    3. The attempt at a solution

    I know that I can solve this problem by introducing f:=x->ln(1+x)-x and studying f, but I'd like to use x->ln(1+x)'s concavity so I tried to find lambda, x and y in :
    ln(1+(lambda)*x+(1-lambda)*y)>=(lambda)*ln(1+x)+(1-lambda)*ln(1+y)...


    Any help is welcome...
     
  2. jcsd
  3. Jan 4, 2010 #2
    Re: ln(1+x)<=x

    You are trying too hard.

    Do you know the taylor series expansion for e^x? That is all you need.
     
  4. Jan 4, 2010 #3
    Re: ln(1+x)<=x

    I find your solution quite interesting too (I had not thought about it) but I'd like to find a solution using the concavity of this function x:=->ln(1+x) (though I admit it may be too hard for that question).
     
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