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Math Help - Prove the series is diverge

  1. #1
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    Prove the series is diverge

    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.


    Give examples to show that if \sum a_k and \sum b_k both diverge, then each of the series

    \sum (a_k + b_k) and \sum (a_k - b_k)

    may converge or may diverge
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  2. #2
    Senior Member Tinyboss's Avatar
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    Quote Originally Posted by wopashui View Post
    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.
    Suppose \sum (a_k+b_k) converges to L and \sum a_k converges to K. For any \varepsilon>0, there exist integers N, M so that \left|\sum_{k=1}^n(a_k+b_k)-L\right|<\varepsilon/2 whenever n>N and \left|\sum_{k=1}^na_k-K\right|<\varepsilon/2 whenever n>M. Then since |x-y|\le|x|+|y|, we have \left|\sum_{k=1}^nb_k-(L-K)\right|<\varepsilon whenever n>max(N,M), contradicting the divergence of that series.

    Quote Originally Posted by wopashui View Post
    Give examples to show that if \sum a_k and \sum b_k both diverge, then each of the series

    \sum (a_k + b_k) and \sum (a_k - b_k)

    may converge or may diverge
    Just define b_k=a_k=1 to make \sum (a_k + b_k) diverge and \sum (a_k - b_k) converge, and b_k=-a_k=1 to get it the other way around.
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