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Use the definition of convergence to prove that the lim (1/2)^n=0

  1. Oct 13, 2011 #1
    1. The problem statement, all variables and given/known data

    Use the definition of convergence to prove that lim n→∞ (1/2)^n=0

    The definition of convergence says |a_n-L|<ε


    2. Relevant equations



    3. The attempt at a solution

    As I understand it:

    |(1/2)^n-0|<ε

    |(1/2)|^n<ε

    then I need to solve for n?

    n>(ln(ε))/ln(|1/2|)

    Then I choose N=(ln(ε))/ln(|1/2|) but I don't understand why.

    n>N>(ln(ε))/ln(|1/2|)??

    Given that ε>0 and n>N

    then n>(ln(ε))/ln(|1/2|)

    then solving for ε I get the statement

    |(1/2)|^n<ε

    from above. Thanks!
     
  2. jcsd
  3. Oct 13, 2011 #2

    Mark44

    Staff: Mentor

    The definition actually says quite a bit more than this, in part about at what point in the sequence this inequality is true.
    You don't need the absolute values, since 1/2 and (1/2)n are positive for all positive integers n.
    Let's back up a bit.
    You want to find a number N so that for a given ε > 0 and any n >= N, (1/2)n < ε.
    Take ln of both sides: n ln(1/2) < ln(ε)
    Divide both sides by ln(1/2), which is a negative number.
    n > ln(ε)/ln(1/2)

    The direction of the inequality changed because we divided by a negative number, ln(ε).

    Note that ε is typically a very small (i.e., much less than 1), but positive number, so ln(ε) < 0, which means that ln(ε)/ln(1/2) > 0. The smaller ε is, the larger this expression is.

    Take N to be the next largest integer that is greater than ln(ε)/ln(1/2). Then for any number n >= N, (1/2)n < ε.

    To see how this works it might be helpful to actually pick a number for ε, say ε = 0.01. Go through the same process as above to find an index N for which all of the terms in the sequence {(1/2)n} are smaller than ε.





     
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