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Improper Integrals and Series (convergence and divergence)

  1. May 23, 2013 #1
    Is it safe to say when an integral has an infinite boundary [itex]\int_n^∞ a_{n}[/itex] and the limit yields a finite number, then the integral is said to converge.

    And when a series has an upper limit of infinity [itex]\sum_n^{∞}a_{n}[/itex] and the limit yields a finite number, then the series is said to diverge.
     
    Last edited: May 23, 2013
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  3. May 24, 2013 #2

    SteamKing

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    If the limit of an infinite series yields a finite number, the series converges. Why would you say opposite things about series and integrals?
     
  4. May 29, 2013 #3
    Oh, I made a conceptual mistake. I was thinking that if the limit of an infinite series yields a finite number, it meant that the terms of the series would "taper off" at some constant term. And that this constant term would be added up indefinitely, thus ∞ = divergent. For instance, if the limit of an infinite sequence yielded the finite number, 1/4, this meant:


    a_{1}+ a_{2} + a_{3} + a_{4} + a_{5}.....1/4 + 1/4 + 1/4 + 1/4 + 1/4.... = ∞ , thus divergent


    I understand now that the finite number, 1/4, in the limit of an infinite series means, that the actual sum of the series is a finite number, thus converges.

    a_{1}+ a_{2} + a_{3} + a_{4} + a_{5}..... = 1/4 ,thus convergent.

    Thanks for the correction.
     
  5. May 30, 2013 #4

    HallsofIvy

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    Note that "yielding an infinite number" is not the only way a series can diverge.
    The series [itex]\sum_{n= 0}^\infty (-1)^n[/itex] is also divergent.
     
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