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Math Help - Sequences, convergence and divergence.

  1. #1
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    Sequences, convergence and divergence.

    Let { {a_n}} be a sequence and k be a positive integer.

    a) If { {a_n}} converges to L, what are the limits of { {a_{99+n}}} and { {a_{k+n}}}? Why?

    b) If { {a_n}} diverges, what is the limit of { {a_{k+n}}}? Why?
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  2. #2
    Super Member Matt Westwood's Avatar
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    There's a theorem somewhere to the effect that if a sequence tends to a limit, then any subsequence of that sequence tends to the same limit. Let me go away and find it ...

    Aha, found it:

    http://www.proofwiki.org/wiki/Limit_of_a_Subsequence

    ... work in progress to expand it to make it more general, but it's there for the real number plane.
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    i read this over and over and i still have no idea how to do this question haha
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    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by qzno View Post
    i read this over and over and i still have no idea how to do this question haha
    May be the following fill be helpful.
    Just think that you walk from the point A to the point B.
    And suppose that there is no other path is possible, i.e., you go on a unique path, and while walking mark your steps with numbers, step 1, step 2, and so on.
    Say you have reached to B after 100 steps.
    Now, can you tell me, could you reach at B without making steps 1,2,3,4,5 if you had chance to start from the point that you made step 6?
    Or if you had enough long legs, could you jump from the place of step 2 to the place of step 4, then to the place of step 6, and so on, to reach B?

    What you wrote above is exactly the same, a_{n}\to L as n\to\infty, and what happens if I ignore the first 99 terms? I will still be able to reach to L by running on the remaining terms of the sequence...

    I think it is now more clear, let me know if not...
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  5. #5
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    So your saying that I would get the same answer from an expression marked { {a_n}} as i would an expression { {a_{99+n}}}?
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  6. #6
    Senior Member bkarpuz's Avatar
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    Wink

    Quote Originally Posted by qzno View Post
    So your saying that I would get the same answer from an expression marked { {a_n}} as i would an expression { {a_{99+n}}}?
    of course...
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