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Convergence of series

  1. Sep 20, 2009 #1
    I have an infinite series, let's say Σan, and I can split it into Σbn + Σcn. If one of the series is convergent (let's say Σbn) while the other is divergent (Σcn), is it safe to say that the original series (Σan) is also divergent?

    If to show an example, I have:
    [tex]\sum {\frac{1+n}{1+n^2}} = \sum {\frac{1}{1+n^2}} + \sum {\frac{n}{1+n^2}}[/tex]
    Intuitively, I can say that yes, but is this enough?
    Or maybe just for cases where both of the series are either negative or positive (positive in this case)?
     
  2. jcsd
  3. Sep 20, 2009 #2

    Office_Shredder

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    If

    Σan = Σbn + Σcn

    where the b series is convergent and the c series is divergent, you can think of it like this:

    Σan - Σbn = Σcn

    And if the series Σan is convergent, then you can combine that with the series Σbn) and get that Σcn is convergent. But it's not, so Σan must be divergent
     
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