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Math Help - analysis help

  1. #1
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    analysis help

    for positive real numbers x how to show that

    1/x+1 < or = ln(x+1)/x < or = to 1
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  2. #2
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    Re: analysis help

    Quote Originally Posted by sbacon1991 View Post
    for positive real numbers x how to show that
    1/x+1 < or = ln(x+1)/x < or = to 1
    Napier"s inequality is:
    If 0<a<b then \frac{1}{b}\le\frac{\ln(b)-\ln(a)}{b-a}\le\frac{1}{a}.

    That is proved using the mean value theorem.

    Let b=x+1~\&~a=1.
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  3. #3
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    Re: analysis help

    Show it using our de nition of ln(x) as intergral between 1 and x of 1/t dt. do not use any other defination. and how could you draw a diagram of this
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  4. #4
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    Re: analysis help

    Quote Originally Posted by sbacon1991 View Post
    Show it using our de nition of ln(x) as intergral between 1 and x of 1/t dt. do not use any other defination. and how could you draw a diagram of this
    Why did you not say that in the first place.
    Let x>0 and \ln (x) = \int_1^x {\frac{1}{t}dt}
    Now clearly \frac{d\ln(x)}{dx}=\frac{1}{x}.
    Use the mean value theorem to prove the Napier inequality.
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  5. #5
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    Re: analysis help

    what do you mean by uthe sing mean value theorem to prove the Napier inequality?
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  6. #6
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    Re: analysis help

    Quote Originally Posted by sbacon1991 View Post
    what do you mean by uthe sing mean value theorem to prove the Napier inequality?
    Do you know what the mean value theorem says?
    You posted this in the analysis forum.
    Therefore, we can safely assume that something as basic as that theorem can be used. If not, what level is this question on?
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  7. #7
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    Re: analysis help

    its a university question and is there any way to draw a graph for the answer?
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