# Thread: Conditional convergence

1. ## 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?

2. 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 $\displaystyle a_1,a_2,...$ be the positive terms (there have to be infinitely many of them). Let $\displaystyle b_1,b_2,...$ be the negative terms (there have to be infinitely many of them). Also, $\displaystyle a_1+a_2+.... = \infty$ and $\displaystyle b_1+b_2+... = -\infty$ (because otherwise we have absolute convergence). Pick enough terms so that $\displaystyle a_1+...+a_{k_1}> 1$. Then pick $\displaystyle b_1$. Then pick enough terms so that $\displaystyle a_1+...+a_{k_1}+b_1+a_{k_1+1}+...+a_{k_2} >2$. Then pick $\displaystyle b_2$. And keep on going.