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Thread: Sequences, convergence and divergence.

  1. #1
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    Sequences, convergence and divergence.

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

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

    b) If {$\displaystyle {a_n}$} diverges, what is the limit of {$\displaystyle {a_{k+n}}$}? Why?
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  2. #2
    MHF Contributor 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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  4. #4
    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 $\displaystyle A$ to the point $\displaystyle B$.
    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 $\displaystyle 1$, step $\displaystyle 2$, and so on.
    Say you have reached to $\displaystyle B$ after $\displaystyle 100$ steps.
    Now, can you tell me, could you reach at $\displaystyle B$ without making steps $\displaystyle 1,2,3,4,5$ if you had chance to start from the point that you made step $\displaystyle 6$?
    Or if you had enough long legs, could you jump from the place of step $\displaystyle 2$ to the place of step $\displaystyle 4$, then to the place of step $\displaystyle 6$, and so on, to reach $\displaystyle B$?

    What you wrote above is exactly the same, $\displaystyle a_{n}\to L$ as $\displaystyle n\to\infty$, and what happens if I ignore the first $\displaystyle 99$ terms? I will still be able to reach to $\displaystyle L$ 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 {$\displaystyle {a_n}$} as i would an expression {$\displaystyle {a_{99+n}}$}?
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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 {$\displaystyle {a_n}$} as i would an expression {$\displaystyle {a_{99+n}}$}?
    of course...
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