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Math Help - Calculus proof help

  1. #1
    Junior Member
    Joined
    Jan 2009
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    Calculus proof help

    Hi! I have a midterm on Monday which will be based on homework, and there is ONE problem that I am having difficulty answering.

    Which is..

    Use the following equation: * <= * means greater than or equal to

    e^(1/(n+1)) <= 1 + 1/n <= e^(1/n)

    to prove that for n>0,

    (1 + 1/n)^n <= e <= (1 + 1/n)^(n+1)

    This is part of the section in my calculus book called "Compound Interest and Present Value"



    Thanks!
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
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    France
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    1,458
    Hi

    Starting from
    e^{\frac{1}{n+1}} \leq 1+\frac{1}{n} \leq e^{\frac{1}{n}}

    Take the natural logrithm (which is an increasing function therefore does not change the sense of inequalities)
    \frac{1}{n+1} \leq ln\left(1+\frac{1}{n}\right) \leq \frac{1}{n} (1)

    Now multiply (1) by n
    n\: ln\left(1+\frac{1}{n}\right) \leq 1

    ln\left({1+\frac{1}{n}}\right)^n \leq 1

    And apply exponentiation (which is an increasing function therefore does not change the sense of inequalities)

    \left({1+\frac{1}{n}}\right)^n \leq e

    Multiply (1) by (n+1) instead of n to get the second part of the inequality you are looking for
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  3. #3
    Junior Member
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    Thank you so much!
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