Verified answer

I think your question isn't complete, something must have happened when you posted it here.

Anyway, I'm supposing the series we have to analyze is Sum(a_k+b_k), right?

So, you say that a series is absolutely convergent when Sum(|a_k|) for k from zero to infinite is less than infinite ( |a_k| is the absolute value of a_k ). We have that:

Sum(|a_k|) < infinite

Sum(|b_k|) < infinite

Now let's look at the other series:

Sum( |a_k + b_k| ) <= Sum( |a_k|+|b_k| ) = Sum( |a_k| ) + Sum( |b_k| ) < infinite

So the series is absolutely convergent. The first step hold because |a+b| <= |a| + |b| for any a and b.

In general the converse does not hold. Here is a counter-example: suppose b_k = -a_k, and that Sum(a_k) is not absolutely convergent. But Sum(|a_k+b_k|) = 0, thus it is absolutely convergent, and Sum(a_k) is not, so the converse does not hold.