Results 1 to 2 of 2

Thread: Prove the series is diverge

  1. #1
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Prove the series is diverge

    Show that if the series $\displaystyle \sum a_k$ converges and the series $\displaystyle \sum b_k$ diverges, then the series $\displaystyle \sum (a_k+b_k)$ diverges.


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

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

    may converge or may diverge
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    Quote Originally Posted by wopashui View Post
    Show that if the series $\displaystyle \sum a_k$ converges and the series $\displaystyle \sum b_k$ diverges, then the series $\displaystyle \sum (a_k+b_k)$ diverges.
    Suppose $\displaystyle \sum (a_k+b_k)$ converges to L and $\displaystyle \sum a_k$ converges to K. For any $\displaystyle \varepsilon>0$, there exist integers N, M so that $\displaystyle \left|\sum_{k=1}^n(a_k+b_k)-L\right|<\varepsilon/2$ whenever n>N and $\displaystyle \left|\sum_{k=1}^na_k-K\right|<\varepsilon/2$ whenever n>M. Then since $\displaystyle |x-y|\le|x|+|y|$, we have $\displaystyle \left|\sum_{k=1}^nb_k-(L-K)\right|<\varepsilon$ whenever n>max(N,M), contradicting the divergence of that series.

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

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

    may converge or may diverge
    Just define $\displaystyle b_k=a_k=1$ to make $\displaystyle \sum (a_k + b_k)$ diverge and $\displaystyle \sum (a_k - b_k)$ converge, and $\displaystyle b_k=-a_k=1$ to get it the other way around.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Series: Converge of Diverge
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 6th 2010, 04:05 AM
  2. Prove Sequence is Converges or diverge
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 17th 2010, 07:26 PM
  3. Does this series converge or diverge?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Oct 30th 2009, 07:57 AM
  4. Prove that harmonic series diverge
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Aug 9th 2009, 02:22 AM
  5. Series Converge or diverge 2
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 23rd 2008, 06:18 PM

Search Tags


/mathhelpforum @mathhelpforum