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Infinite Series (2 diverge -> 1 converge)

  1. Oct 29, 2007 #1
    Infinite Series (2 diverge --> 1 converge)

    I've been trying to figure this question out:

    Find examples of two positive and decreasing series, [tex]\sum a_n[/tex] and [tex]\sum b_n [/tex], both of which diverge, but for which [tex]\sum min(a_n,b_n)[/tex] converges.

    It doesn't make any sense to me that any positive and decreasing divergent series can be combined with another to produce a convergent series. Thanks in advance.
     
  2. jcsd
  3. Oct 29, 2007 #2

    Zurtex

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    Are you sure you don't mean 2 positive and decreasing sequences an and bn such that [itex]\sum a_n[/itex] and [itex]\sum b_n[/itex] diverge?

    Just asking before I give it a crack.
     
  4. Oct 29, 2007 #3
    100% sure.
     
  5. Oct 29, 2007 #4

    Zurtex

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    Off the top of my head I'd say the question is flawed. If:

    [tex]\sum_{n=0}^{i} a_n[/tex]

    Is a positive, decreasing divergent series in i, then WLOG we can say that a0 >> 0. We can also say that an<0 for all n > 0. So if we take bn = -an and look at this sequence:

    [tex]\sum_{n=1}^{i} b_n[/tex]

    All summation terms are positive, the series doesn't converge and the series is strictly less than infinity. Now if the series is strictly less than infinity it is necessary that:

    [tex]\lim_{n \rightarrow \infty} b_n = 0[/tex]

    And that for some n > N we have that:

    [tex]\sum_{n=N}^{\infty} b_n < B_0 \quad B_0 \in \mathbb{R}[/tex]

    But because bn > 0 and the real numbers are complete, there must in fact be some B such that:

    [tex]\sum_{n=N}^{\infty} b_n = B \quad B \in \mathbb{R}[/tex]

    (All the rigorous proof words escape my mind right now, but I'm quite confident this holds, it's reminding me of some work I did on my metric spaces course).
     
    Last edited: Oct 29, 2007
  6. Oct 29, 2007 #5
    thats not the first time i've come across a flawed question like this.
    ...thanks anyway.
     
  7. Oct 29, 2007 #6

    morphism

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    What is a positive decreasing series?

    I'm pretty certain that you mean a_n and b_n are decreasing sequences. (In which case the result is true.)
     
  8. Oct 30, 2007 #7
    thats what i'm thinking. i am actually using the first version of a book that is now in its 8th revision, so i'm guessing there are a few errors. however i saw similar question in another book which did specify a_n and b_n being positive & decreasing sequences, while their series were divergent. I also need some advice on that problem.

    Thanks.
     
    Last edited: Oct 30, 2007
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