Is it possible for a series of positive terms to converge cond...

Question

Is it possible for a series of positive terms to converge conditionally? Explain.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says suppose the sum of C sub N is a series where each CN is greater than 0. Which statement is true? And we're given 4 possible choices as our answers. For choice A, we have it can converge conditionally. For choice B, we have it can only diverge. For choice C we have if it converges, it must converge absolutely, and for choice D, it can never converge absolutely. So what we're going to do is look at each statement individually and determine whether they are true or false. So starting with choice A, we have it can converge conditionally. So we call for a series to converge conditionally that the sum of CN must converge, but the sum of the absolute value of CN is going to diverge. But if we look at our case, where CN is greater than 0 for each CN, that means that the sum of the absolute value of CN. It's going to be equal to the sum of CN. Now since CN is positive, greater than 0, that means that we can drop the absolute value signs around the absolute value of CN and we'll get back just CN. So this means that if the sum of CAN converges, it's already absolutely convergent and cannot be conditionally convergent. So that means that choice A is false. Now, if you look at choice B, it says that it can only diverge. And this statement is also false because we can provide a counterexample. And the counterexample that we're going to look at is going to be the sum. Of one divided by. 2 raise to the power in. And this series converges. Since 1 half is less than 1. And every term in our series here is a positive term. And so since the series converges, that means that it provides as a counterexample, and that means that the sum of C of N could potentially converge. Now, if we look at choice C, it says that if it converges, it must converge absolutely. And we've already discussed this when we were looking at choice A. We saw that that if the sum of CN converges, that means that the sum of the absolute value of CNN converges. And that is the definition of absolute convergence. So that means that B or C is true. And if we look at choice D, It says that it can never converge absolutely. Well, we just showed that it could possibly converge absolutely if the original series converges. So that means that choice D is false as well. So that means that the only true statement is choice C, and that's going to be the answer for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Is it possible for a series of positive terms to converge conditionally? Explain.

Key Concepts

Positive Term Series

Absolute Convergence

Conditional Convergence

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