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Australian HSC maths extension 2 test question

  1. Oct 27, 2009 #1

    Mentallic

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    1. The problem statement, all variables and given/known data
    This question is from the Australian HSC maths extension 2 test. Q8b)

    Let n be a positive integer greater than 1.

    The area of the region under the curve y=1/x from x=n-1 to x=n is between the areas of two rectangles.

    Show that [tex]e^{-\frac{n}{n-1}}<\left(1-\frac{1}{n}\right)^n<e^{-1}[/tex]


    3. The attempt at a solution

    The area under the curve is more than the smaller rectangle but less than the larger rectangle.

    Thus, [tex]\frac{1}{n}<\int^n_{n-1}\frac{dx}{x}<\frac{1}{n-1}[/tex]

    After manipulating somewhat:

    [tex]\frac{1}{n}<ln\left(\frac{n}{n-1}\right)<\frac{1}{n-1}[/tex]

    [tex]e^{\frac{1}{n}}<\frac{n}{n-1}<e^{\frac{1}{n-1}}[/tex]

    but I'm unsure how to get to the answer...
     
  2. jcsd
  3. Oct 27, 2009 #2

    lanedance

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    Re: Inequalities

    try inverting everything as a next step, then consider raising everything to a power
     
  4. Oct 27, 2009 #3

    Mentallic

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    Re: Inequalities

    The reciprocal... how could I miss that... thanks lanedance.
     
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