Determine the convergence or divergence of the series. ∑ k = 1 ∞ ( − 1 ) k 4 3 k + 2 \sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}4}{3k+2}

this problem, we are asked to determine the convergence or divergence of the given series. So we have the sum from k equals one to infinity of negative one to the k power times four over three k plus two. Now this is kind of interesting, because notice here that we're given a series, but we're not told what type of series it is, nor are we told what tests specifically to do. So all of this we have to figure out ourselves. So let's see how we can go about doing this. I went ahead and copied over this flowchart right here that we saw, and this will help us to determine which test we need to use. Now a good place to start when we're not sure where is to start with the divergence test and see if that's gonna give us some kind of answer. Well, for the divergence test, what I want to do is take the limit as, in this case, it's going to be k approaches infinity since we're dealing with k instead of n, and it's going to be of this whole piece right here. I'm going to call this whole thing a sub k. Now what's gonna happen as k approaches infinity here? Well, let's see that we have k at two points of this fraction, and notice that the k on top is this is just an exponent raised on negative one. Negative one raised to an exponent of k is just gonna cause us to alternate between positive one, negative one, positive one, negative one, or in this case, negative one, positive one, negative one, positive one. So this is just gonna constantly go back and forth between one and negative one. Now this k right here is is in the denominator. It's multiplied by three, and as this blows up to infinity, this is gonna cause the whole denominator to blow up and get really, really big. So that means this whole thing is eventually just gonna go to zero, which means the divergence test would be inconclusive. So we're gonna need to use some other test here. Well, next, I'll take a look and see if this is some kind of special series. Now looking at this, I don't really see a case where this is a geometric series. There's no way we can really manipulate this fortune to fit the look of a geometric series, so it's not gonna be that. I can also see we don't have a simple case where it's a harmonic or p series. It almost looks like it could be a harmonic series with this three k, but since we have this negative one to the k on top, it's not a harmonic series, so this is not gonna be a special case. Now is this series going to be alternating? Well, this is interesting because recall that we just talked about how we have this negative one to the k power that will cause this to alternate between negative and positive, negative, positive. So because of that, this actually is alternating. So I'm going to use the alternating series test to determine whether this series converges or diverges. Now recall we have a rule for the alternating series test. There are two criteria that need to be met for the series we have. And since we can see this is the part that is making our series alternate, then our a sub k is going to be everything else that we have. So it's going to be this four over three k plus two. Now first off, I need to figure out whether the limit as k approaches infinity of a sub k is going to go to zero. Now looking at this, since I see we have this k here in the denominator, this is just gonna blow up to infinity, causing the whole fraction to go to zero. So, yes, this is going to come out to zero, meaning the first condition is indeed met. Now let's check our second condition. We need to make sure this is going to be decreasing. And to check whether or not this is decreasing, well, what I need to do is look at the next term up. So you can see this here is my ace of k. And what I'm going to do is check the next term up, ace of k plus one. And to find this, well, I can see this top here is just gonna stay the same, but on bottom, this k is gonna be replaced with k plus one. Now you clearly see the denominator that we have right here is a bigger overall denominator than what we had before, and a bigger denominator equates to a smaller overall value refraction. So because of that, it's safe to say that this a sub k plus one is less than the a sub k, meaning we're decreasing as we go up by each step. So because of that, we can say that the second criteria is also met where we have to have a sub k decreasing. So because we could see that was the case, this condition is also met. And because both these conditions are met, we can say that this series converges. So this right here will be the solution to the problem, and that has you is how you can take a series like we have right here and figure out what tests you need to apply, what type of series it is, and and how we can determine its convergence just starting from scratch with a random series like this. So let's get some more practice from here. Hope you found this helpful. See you in the next video.

14. Sequences & Series

Convergence Tests

Calculus

14. Sequences & Series

Convergence Tests

Struggling with Calculus?

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Multiple Choice

Determine the convergence or divergence of the series.
$\sum_{k = 1}^{\infty} \frac{{(- 1)}^{k} 4}{3 k + 2}$

Inconclusive

Diverges

Converges

Verified step by step guidance

Step 1: Recognize that the series ∑k=1∞(−1)^k * 4 / (3k + 2) is an alternating series because of the term (-1)^k, which alternates the sign of each term.

Step 2: Apply the Alternating Series Test (Leibniz Test). This test states that an alternating series converges if: (a) the absolute value of the terms decreases monotonically, and (b) the limit of the terms as k approaches infinity is zero.

Step 3: Check condition (a): Analyze the term |4 / (3k + 2)|. Observe that as k increases, the denominator (3k + 2) grows, causing the term to decrease monotonically.

Step 4: Check condition (b): Compute the limit of the term as k approaches infinity. Evaluate lim(k→∞) 4 / (3k + 2). Since the denominator grows without bound, the limit is 0, satisfying the second condition.

Step 5: Conclude that both conditions of the Alternating Series Test are satisfied. Therefore, the series converges.

5:44m

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Struggling with Calculus?

Determine the convergence or divergence of the series. ∑k=1∞(−1)k43k+2\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}4}{3k+2}

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Convergence Tests practice set

Determine the convergence or divergence of the series.
$\sum_{k = 1}^{\infty} \frac{{(- 1)}^{k} 4}{3 k + 2}$