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Inequality help

  1. Oct 23, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that

    [tex]\frac{1}{2^k+1}+\frac{1}{2^k+2}+...+\frac{1}{2^{k+1}}>\frac{1}{2}[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I cannot figure this out. It is part of a larger proof that I am trying to understand. Any help would be appreciated!
     
  2. jcsd
  3. Oct 23, 2008 #2
    Each of the terms [tex]\frac{1}{2^k+1},... \frac{1}{2^{k+1}-1} [/tex] is larger than [tex]\frac{1}{2^{k+1}} [/tex]

    There are 2*2^k - 2^k = 2^k terms so the whole sum is certainly larger than 1/2:

    [tex]\frac{1}{2^k+1}+...+ \frac{1}{2^{k+1}} > \frac{2^k}{2^{k+1}} = \frac{1}{2}[/tex]
     
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