Can a series of nonnegative numbers converge conditionally?

[happyg1]

Hi, We're debating the question "Can a series of nonnegative numbers converge conditionally?"
I say no because if all of the terms are nonnegative then they are the same as their absolute values. My classmate disagrees and says that there is a series that has nonegative terms whose absolute value diverges. I'm really confused. He won't tell us what this divergent series is and I can't come up with a counterexample of my own. I keep staring at the definition of absolute convergence and getting more confused.
help me please.
CC

 

[Muzza]

You are correct, your classmate is wrong.

Suppose a_n >= 0 and that $\sum_{k = 1}^{\infty} a_k$ converges, and $\sum_{k = 1}^{\infty} |a_k|$ diverges. But |a_n| = a_n, so that $\sum_{k = 1}^{\infty} |a_k| = \sum_{k = 1}^{\infty} a_k$, so that that series both converges and diverges. Clearly nonsense.

My classmate disagrees and says that there is a series that has nonegative terms whose absolute value diverges.

Of course there is such a series (consider $\sum_{k = 1}^{\infty} |k|$), but that's totally irrelevant.

 

[happyg1]

Thanks for clearing that up. That's exactly what I thought!