Conceptual question regarding sum of series

[fk378]

Homework Statement

If the sum(a_n) and the sum(b_n) are both divergent, is the sum(a_n + b_n) necessarily divergent?

The Attempt at a Solution

At first I thought it must be divergent, but then I asked, what if a_n is 1/n and b_n is -1/n... then their sum would be 0.

Does this logic make sense? And if so, would it imply that sum(a_n + b_n)=0, and is therefore convergent?

 

[Gib Z]

Your logic is correct. We can even get simpler examples. a_n = n, b_n = -n. Each diverges individually, but a_n + b_n = 0 for every term. And yes, it implies the sum of (a_n + b_n) is 0, convergent.