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Math Help - sequence question

  1. #1
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    sequence question

    Let (an) be a sequence such that lim as n approaches infinity of (an + an+1) exists and lim as n approaches infinity of (anan+1) exists.

    I'm pretty sure that lim as n approaches infinity of (an - an+1) does not exist (the textbook seems to imply this), yet I can't think of a counterexample. Can someone give me a sequence where this limit does not exist?
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  2. #2
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    Have you studied the concept of Cauchy Sequences?
    If you have, then you know that if (a_n) converges then (a_n) is a Cauchy sequence.

    Being a Cauchy sequence what does that tell you about the difference \left| {a_n  - a_{n + 1} } \right|?
    In other words, does it mean that {\left( {a_n  - a_{n + 1} } \right)} converges?
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  3. #3
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    Thanks for answering, but no we haven't studied those yet.

    Are you saying it does converge?
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  4. #4
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    Quote Originally Posted by CindyMichelle View Post
    Thanks for answering, but no we haven't studied those yet.
    Well then here is another way to think about it.
    If (a_n) converges then almost all of its terms get very close to the limit value. Does that make sense? Therefore almost all of its terms have a difference close to zero. So {\left( {a_n  - a_{n + 1} } \right)} converges to zero.
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  5. #5
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    How about an = (-1)^(n). Then an+1 = (-1)^(n+1).
    limit as n approaches inf of abs(an - an+1) = 2.
    limit as n approaches (an - an+1) does not exist.
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  6. #6
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    Does this not contradict what you are saying?
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