determine convergence, absolute or conditional

**wopashui** · April 8th 2011, 01:24 PM

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

**Plato** · April 8th 2011, 03:10 PM

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 $|a_nc_n|~?$

**wopashui** · April 9th 2011, 07:50 PM

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?

**Plato** · April 10th 2011, 03:32 AM

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.

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