Thread: Conditional convergence

I'm familiar with the theorem that states a conditionally convergent series can be rearranged to converge to any real number or diverge. I've seen how to prove convergence to any finite value, but how would I show divergence? Specifically, how can I show that any conditionally convergent series can be rearranged so that it diverges to infinity?

Originally Posted by spoon737

I'm familiar with the theorem that states a conditionally convergent series can be rearranged to converge to any real number or diverge. I've seen how to prove convergence to any finite value, but how would I show divergence? Specifically, how can I show that any conditionally convergent series can be rearranged so that it diverges to infinity?

Maybe this works. Let be the positive terms (there have to be infinitely many of them). Let be the negative terms (there have to be infinitely many of them). Also, and (because otherwise we have absolute convergence). Pick enough terms so that . Then pick . Then pick enough terms so that . Then pick . And keep on going.