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Homework Help: Prove these inequalities

  1. Jan 25, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that
    a.) (1-(1/n2))n > 1- 1/n

    b.) (1+ 1/(n-1))n-1 < (1 + 1/n)n

    when n=2,3,4,5,...

    2. Relevant equations
    Bernoulli's inequality
    (1+x)n ≥ 1+nx,
    when x ≥-1 and n=2,3,4,5,...

    (1+x)n >1+nx,
    when x ≥-1, x≠0 and n=2,3,4,5,..

    3. The attempt at a solution
    I applied Bernoulli's inequality.

    First I checked 'the requirements'.
    -1/n2 > -1 because n=1,2,3,... and -1/n2 ≠ 0 OK

    Then (1-(1/n2))n > 1+ (- 1/n2)*n=1- 1/n Ok, done.

    b.) I think I am supposed to apply Bernoulli's inequality as in part a, but don't have an idea how to get started.
    Last edited: Jan 25, 2016
  2. jcsd
  3. Jan 25, 2016 #2


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    What do you know about ##\displaystyle\ \frac{1}{n-1} \ ? ##
  4. Jan 25, 2016 #3
    Could you please elaborate?
  5. Jan 25, 2016 #4


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    Is it positive?

    Does it have an upper bound ?
  6. Jan 25, 2016 #5
    It is always positive.
    0 < 1/(n-1) ≤ 1
    And it does have an upper bound.
    What's the next step?
    Last edited: Jan 25, 2016
  7. Jan 25, 2016 #6

    Ray Vickson

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    We are not allowed to tell you that; PF rules require you to do the work. However, if you get stuck at some point, we are permitted to give hints, but first you need to reach the point of getting stuck on your own.
  8. Jan 26, 2016 #7
    I have read the forum rules. I did the work at part a and now I am stuck at part b.

    0 < 1/(n-1) ≤ 1 But how that will help?
  9. Jan 26, 2016 #8
  10. Jan 27, 2016 #9
    Nevermind. Now I figured it out.
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