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Math Help - Conditional convergence

  1. #1
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    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?
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    Quote Originally Posted by spoon737 View Post
    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 a_1,a_2,... be the positive terms (there have to be infinitely many of them). Let b_1,b_2,... be the negative terms (there have to be infinitely many of them). Also, a_1+a_2+.... = \infty and b_1+b_2+... = -\infty (because otherwise we have absolute convergence). Pick enough terms so that a_1+...+a_{k_1}> 1. Then pick b_1. Then pick enough terms so that a_1+...+a_{k_1}+b_1+a_{k_1+1}+...+a_{k_2} >2. Then pick b_2. And keep on going.
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