Assume $ \sum_{n=1}^{\infty}a_{n} $ is absolutely convergent.
and assume $ \sum_{n=1}^{\infty}b_{n}$ is conditionally convergent.
What can we say for sure about $ \sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right) $ ?
I'm thinking that for sure, we could say that the series is convergent. But the series will not converge absolutely.
To prove the convergence is not hard I guess, we just use limit arithmetic over the sum sequences aka $ S_{p}=\sum_{n=1}^{p}b_{n},T_{p}=\sum_{n=1}^{p}a_{p} $.
But I'm not sure how to show that the series will not converge absolutely (is it true at all)? because $ |a_{n}+b_{n}|\leq|a_{n}|+|b_{n}| $ we know that $ \sum_{n=1}^{\infty}|b_{n}| $ diverges, but we cannot really say anything based on comparison test.
I'd appreciate some help. Thanks in advance
Best Answer
Use proof by contradiction. Suppose $\sum (a_n+b_n)$ converges absolutely. Then so does $\sum b_n$ because $|b_n| \leq |a_n+b_n|+|a_n|$ which leads to a contradiction.