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Math Help - Show that the sum of a convergent and divergent series is divergent

  1. #1
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    Show that the sum of a convergent and divergent series is divergent

    I am completely stumped on this one, so I could really use a hand with it.

    Show that if the series \sum a_k converges and the series \sum b_k diverges, then the series \sum (a_k+b_k) diverges.
    A rigorous proof is expected, and we are to prove this by contradiction.

    I don't know how I could answer this. This is all I really know that could possibly help me answer it.
    If \sum_{k=0}^{\infty}a_k converges and \sum_{k=0}^{\infty}b_k converges, then \sum_{k=0}^{\infty}(a_k+b_k)
    Moreover, if \sum_{k=0}^{\infty}a_k=L and \sum_{k=0}^{\infty}b_k=M, then \sum_{k=0}^{\infty}(a_k+b_k)=L+M

    I could really use some help with this one. I wasn't able to see my Calculus professor concerning the matter due to an unexplained absence on his part.
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  2. #2
    Super Member General's Avatar
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    Hint: Argue by contradiction.
    Start by assuming that the series \sum (a_k+b_k) converges.
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  3. #3
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    Quote Originally Posted by General View Post
    Hint: Argue by contradiction.
    Start by assuming that the series \sum (a_k+b_k) converges.
    Alright, I've tried your advice out and here's the work I currently have.

    Assume \sum (a_k+b_k) converges.
    Since \sum (-1)a_k=-\sum a_k converges,
    -\sum a_k + \sum (a_k+b_k) = \sum b_k
    This implies that \sum b_k converges. This is a contradiction, since \sum b_k diverges.

    Is this proof correct?
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  4. #4
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    Very good!
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