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Convergence of sequences

  1. Oct 4, 2013 #1

    ppy

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    Hi,

    Let a(n) be a real sequence such that a(n+1)-a(n) tends to zero as n approaches ∞. must a(n) converge? Also an explanation would be great thank you. have been wondering about this
     
  2. jcsd
  3. Oct 4, 2013 #2

    HallsofIvy

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    Staff Emeritus
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    No.

    Let [tex]a_n= \sum_{i=1}^n \frac{1}{i}[/tex].

    Then [itex]a_{n+1}- a_n= 1/(n+1)[/itex] which goes to 0 as n goes to infinity. But the harmonic series does NOT converge so this sequence does not converge.
     
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