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Math Help - Show that ln5 < 1 + 1/2 + 1/3 + 1/4.

  1. #1
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    Show that ln5 < 1 + 1/2 + 1/3 + 1/4.

    Show that ln5 < 1 + 1/2 + 1/3 + 1/4. using integrals

    I guess....First I have to show that the integral from 1 to 5 of \frac {1}x is equal to ln(5).

    Then estimate the integral by a finite sum of left endpoints.

    Since 1/x is decreasing from 1 to 5, the finite sum I get will be strictly greater than the integral.

    But I don't know how to show the steps....

    How do I show that the integral from 1 to 5 of \frac {1}x is equal to ln(5)?

    How do I estimate the integral by a finite sum of left endpoints?
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  2. #2
    MHF Contributor Reckoner's Avatar
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    Quote Originally Posted by cammywhite View Post
    How do I show that the integral from 1 to 5 of \frac {1}x is equal to ln(5)?
    Evaluate \int_1^5\frac{dx}x using the fundamental theorem of calculus.

    How do I estimate the integral by a finite sum of left endpoints?
    Add up the areas of the four rectangles. In each subinterval, \Delta x=1 and f(c)=\frac1c, where x=c is the left endpoint of the subinterval.
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  3. #3
    MHF Contributor Calculus26's Avatar
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    details of proof

    see attachment
    Attached Files Attached Files
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