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Can we say anything on the convergence/or divergence of the series $\sum_{n=1}^{\infty}(a_n+b_n)$ given the convergence/divergence of $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$. What about the converse ? What about if $a_n$ and $b_n$ are all non-negative

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If $\sum_n a_n$ and $\sum_n b_n$ converge, then the sum $\sum_n a_n + b_n$ converges. If $\sum_n a_n$ and $\sum_n b_n$ diverge, then $\sum_n a_n + b_n$ may converge or diverge (just consider adding $1$ to all $a_n$ and adding $-1$ to all $b_n$). Similarly, if $\sum_n a_n + b_n$ converges, then you can't say much about convergence of $\sum_n a_n$ and $\sum_n b_n$, although if one of them converges then the other must also converge. (And if the sum diverges the one of the summand series must diverge).

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  • $\begingroup$ What about if $a_n$ and $b_n$ are all positive ? in that i guess only convergence of both implies convergence of their sum. $\endgroup$ – RIchard Williams Sep 20 '13 at 22:26
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    $\begingroup$ If $a_n$ and $b_n$ are all positive, convergence of the sum is equivalent to both converging, and divergence of the sum equivalent to at least one diverging. This follows basically from that an increasing sequence converges in $\mathbb{R}$ if it's bounded, and diverges to infinity if it's unbounded. $\endgroup$ – Matt Rigby Sep 20 '13 at 22:28
  • $\begingroup$ @user2566092,matt Rigby: "although if one of them converges then the other must also converge" ?. is this true for any arbitrary sequence $\{a_n\}$ and $\{a_n\}$ or only for sequences where one represents the sequence of all positive terms and the other represents the all negative terms. I think that the latter is correct. $\endgroup$ – RIchard Williams Sep 21 '13 at 0:31

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