Proving the convergence of a sequence..

  1. [tex]2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}[/tex]
    etc..
    (the sequence consists only from positive number so the sum is not negative)
    in order to prove that its convergent i need to prove monotonicity and boundedness

    monotonicity:(by induction)

    [tex]a_1=2[/tex]
    [tex]a_2=2.5[/tex]
    so i guess its increasing
    suppose n=k is true:
    [tex]a_{k-1}<a_k[/tex]
    prove n=k+1 ([tex]a_{k}<a_{k+1}[/tex])
    [tex]
    a_k>a_{k-1}\\
    [/tex]
    [tex]
    \frac{1}{a_k}<\frac{1}{a_{k-1}}\\
    [/tex]
    [tex]
    2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\
    [/tex]
    [tex]
    a_{k+1}<a_k
    [/tex]

    i proved the opposite :)
    so this is weird.

    the answer in the book tells me to split the sequence into odd /even sub sequences
    the one is ascending and the other its descending.

    i cant see how many sub sequences i need to split it to
    maybe its 5 or 10
    what is the general way of solving it.
    and how you explained that i proved the opposite
     
    Last edited: Mar 9, 2009
  2. jcsd
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