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Math Help - proving the convergence of a sequence..

  1. #1
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    proving the convergence of a sequence..

    2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}
    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

    monotonicityby induction)

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

    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
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  2. #2
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    Your sequence is a_1  = 2\,\& \,n \geqslant 2,\,a_n  = 2 + \frac{1}{{a_{n - 1} }},
    It is not monotone but it is bounded.
    Hint: look at the odd subscripted terms and then the even.
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  3. #3
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    why its not monotone??

    is it because of my prove
    that i get the opposite result??

    how do you know to split them into two function (on odd index and even index) ??

    there could be endless sub sequences
    ??
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  4. #4
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    Quote Originally Posted by transgalactic View Post
    why its not monotone??
    is it because of my prove IS WRONG
    Here are the first five terms: 2\;,\;\frac{5}{2}\;,\;\frac{12}{5}\;,\;\frac{29}{1  2}\;,\;\frac{70}{29}.
    What are those terms doing?
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  5. #5
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    i understood that they are not increasing nor decreasing.
    so i will just have to check the first few members instead
    of the first two.


    my remaining question is ,how did you decide to split it into two
    sub sequences (by odd index and even index)

    there are infinite number of sub sequences
    why just these two?
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  6. #6
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    Quote Originally Posted by transgalactic View Post
    my remaining question is ,how did you decide to split it into two sub sequences (by odd index and even index)
    The subsequence with odd indices increases and is bounded above by any even indexed term.
    The evens decrease.
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  7. #7
    Member Mentia's Avatar
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    Why cant we say:

     \frac{1 }{x } < 1,  \forall x > 2

     { a}_{ n} > 2, \forall n

    Then,

    2 +  \frac{ 1}{ { a}_{ n}} < 3, \forall n

    So bounded above by 3 and below by 0. This okay?
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  8. #8
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    what did you mean to say here
    "
    why its not monotone??
    is it because of my prove IS WRONG"

    "

    i proved its wrong
    or my prove has a mistake in it?
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