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Numerade Educator



Problem 3 Medium Difficulty

Determine whether the series is absolutely convergent or conditionally convergent.

$ \displaystyle \sum_{n = 0}^{\infty} \frac {( - 1)^n}{5n + 1} $


Conditionally Convergent


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Video Transcript

Let's show that this series is not absolutely conversion so it can do that. We should be looking at the Siri's, obtained from the original one when you take the absolute value of the term. So put that absolute value on there. And let's go ahead and evaluate the absolute value, the denominators positive so it doesn't need absolute value. And the absolute value of negative once at the end is just one. So this is the absolute value. Or, I mean, if we take the absolute value inside the Siri's, this is what we get and this Siri's diverges and many ways to show this, for example, and you could say bye like lemon comparison test. So the way to do this is you can call this, for example, and and then let's define, be and to be won over end. You know that being this is the harmonic series, this diverges Carmen in Siri's diverges. Or you could just use to Pete us with P equals one that also implies that virgins. So let's use the first. We have to show that the hypothesis is satisfied for the limit comparison test. So we look at the limit of a M over BM, So go ahead and plug that in and simplify. You. Get an over five plus one that's one over five and this is inside. It satisfies the following inequality, and this is what allows us to use limit comparison tests. This is important. This is hypothesis can be used unless the limit exists. And it's bigger than zero and less than infinity, not equal to either of those in points. So by Lim comparison tests, since the sum of the being diverges to some of the A and also the bridges. So that justifies why the Siri's is not absolutely conversion. It's all right that oh five in plus one is not absolutely conversion, however, and maybe conditionally will have to go to the next stage. But for the first part, that not that'll be part of our answer going on to the next page. This is where the question is is whether it's conditionally conversion, so this just means that it converges, but not absolutely So this is the question. So we just showed that it's not absolutely conversion, but if we can show that the original savories converges, then by definition it would have been conditionally conversion. So let's write that Siri's back down here, that Siri's that goes from zero to infinity, negative one to the end. Over five and plus one, this thing will converge. And to show it we can use the alternating series test. So to do this test, you look at the absolute value off the end's term here. So I already used the end before. So let me just define it as another letter here. But the book usually will these be in for this instead of CNN? So watch out there when If you compare this to the non station in the book, so there's three conditions to satisfy. One is that we need C N to be positive. This is clearly true because we have positive divided by a positive condition, too. We need the limit of CNN to be zero, and in this case that's clearly true. The denominator goes to infinity. The numerator is just one so that women is zero and the third one is we need that the C N is decreasing. So in other words, going on to the next page, we want C n plus one to be less than or equal to CNN for all in. So in this case, we'LL see n by definition one over five plus one and then what happened on C N plus one. So here you increase and buy one. And this is definitely true, since the denominator on the left is larger. So this shows that the sea and plus one is less than or equal to Seon. So that means that the sequence was also decreasing. So by the alternating Siri's test, the Siri's from zero to infinity convergence and then finally, since this Siri's we just showed a convergence So we can say this, but it does not converge. We also showed this to in the first part of the problem, since it converges, but not absolutely. It's conditionally conversion course, and that's our final answer.