# Thread: determine convergence, absolute or conditional

1. ## determine convergence, absolute or conditional

Suppose that $\displaystyle \sum a_n$ and $\displaystyle \sum b_n$are absolutely convergent series and $\displaystyle \sum c_n$is conditionally convergent. For each of the series $\displaystyle \sum(a_n+b_n)$, $\displaystyle \sum(a_n+c_n)$and $\displaystyle \sum (a_nc_n)$, determine whether is it absolutely convergent or conditionally convergent or neither.

I have already given the answer which are absolutely,conditionally and absolutely, I just need some explaination for $\displaystyle \sum(a_n+c_n)$and $\displaystyle \sum (a_nc_n)$, why is conditionally and absolutely

2. Originally Posted by wopashui
Suppose that $\displaystyle \sum a_n$ and $\displaystyle \sum b_n$are absolutely convergent series and $\displaystyle \sum c_n$ is conditionally convergent. For each of the series $\displaystyle \sum(a_n+b_n)$, $\displaystyle \sum(a_n+c_n)$and $\displaystyle \sum (a_nc_n)$, determine whether is it absolutely convergent or conditionally convergent or neither.
I have already given the answer which are absolutely,conditionally and absolutely, I just need some explaination for $\displaystyle \sum(a_n+c_n)$and $\displaystyle \sum (a_nc_n)$, why is conditionally and absolutely
For $\displaystyle \sum(a_n+c_n)$ looking at the partial sums each series, $A_n~\&~C_n$.
We can rearrange finite sums without effecting the outcome.
Can you give an example that is not absolutely convergent?

Because $\displaystyle \sum c_n$ is conditionally convergent does that not imply that $c_n$ in bounded?
Say that $B>0$ and $|c_n|\le B$ what can you say about $|a_nc_n|~?$

3. Originally Posted by Plato
For $\displaystyle \sum(a_n+c_n)$ looking at the partial sums each series, $A_n~\&~C_n$.
We can rearrange finite sums without effecting the outcome.
Can you give an example that is not absolutely convergent?

Because $\displaystyle \sum c_n$ is conditionally convergent does that not imply that $c_n$ in bounded?
Say that $B>0$ and $|c_n|\le B$ what can you say about $|a_nc_n|~?$

so $|a_nc_n| <=B|a_n|$, and by the comparion test, we know $B|a_n|$ converges absolutely, thus $|a_nc_n|$ converges absolutely. Right?

Does conditional and absolute convergent both imply the series is bounded?

4. Originally Posted by wopashui
so[tex]Does conditional and absolute convergent both imply the series is bounded?
If $\sum {a_n }$ converges then $(a_n)\to 0$.
Any convergent sequence is bounded.