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Chapter 9: Problem 2

Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.

Short Answer

Expert verified
If a series is conditionally convergent, then both the series made from its positive and negative terms are divergent. This is because, if each were convergent, the original conditionally convergent series would also have to be absolutely convergent, which contradicts the given that it's not.

Step by step solution

01

Define the terms

A positive term in a series is a term that is greater than zero, while a negative term is less than zero. We need to form two series from the original one; one containing all the positive terms, let's call this series P. And the other one containing all the negative terms, let's call this series N.
02

Assume conditional convergence

As given, let's say that the original series is conditionally convergent, meaning the sum of the absolute values of all the terms in the series diverges, formally: \[\sum |\a_n|\] diverges.
03

Assumption of divergence for individual series

We want to show that the series of positive terms P diverges and the series of negative terms N diverges. This would mean that if the series P and N were independently summed, each sum would either go off to +鈭�, -鈭�, or never settle on a single finite value.
04

Proof by contradiction

Let's assume, to reach a contradiction, that the series P does converge, say to the number p. Because the series as a whole is conditionally converging, and N = Original Series - P, it should follow that this series too, converges - specifically, to a number minus p. This, however, contradicts the original assumption that the sum of the absolute values of all the terms in the series (\[\sum |\a_n|\]) diverges. Hence, the series P, constituted of the positive terms, should diverge. The same argument can be used to prove the divergence of the series N.

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