 Welcome to the GVSU calculus screencasts. This video is about the alternating series test. Recall that an alternating series is a series whose terms alternate signs between positive and negative. In other words, an alternating series is a series that has the form a sub 1 minus a sub 2 plus a sub 3 minus a sub 4 and so on, where all these terms are non-negative. We typically represent the alternating sign in an alternating series with powers of negative 1, as in the sum k goes from 1 to infinity, negative 1 to the k plus 1 times a sub k. And in the screencast, we're going to apply the alternating series test to determine the convergence or divergence of a few series. As an example, consider the series sum k goes from 1 to infinity minus 1 to the k plus 1, 6 over 3 k plus 2. Recall that the alternating series test tells us that an alternating series will converge if and only if the sequence of non-negative terms, the sequence a sub k, decreases to 0. So pause the video for a moment and look at our example of the alternating series, negative 1 to the k plus 1, 6 over 3 k plus 2, and explicitly identify the a sub k, and then determine if the alternating series test applies to this series, and if so, what conclusions can we draw? Then resume the video when you're ready. Writing out the first few terms of this series, we see that when k equals 1, we have 6 fifths, when k is 2, we have negative 6 eighths, when k is 3, we have 6 elevenths, when k is 4, we have negative 6 thirteenths. So this is an alternating series, and in this case, a sub k is 6 over 3 k plus 2. Now we need to know if the sequence a sub k decreases to 0. Now, k plus 1 is bigger than k, so by inverting both sides, we can see that a sub k, which is 6 over 3 k plus 2, has to be bigger than 6 over 3 times the quantity k plus 1 plus 2, which is a sub k plus 1 for each k. So the sequence a sub k decreases. We also know that the limit as k goes to infinity of the sequence 6 over 3 k plus 2 equals 0, so that means that the sequence a sub k decreases to 0. An alternative way to show that the sequence decreases is to consider the continuous function f of x equals 6 over 3 x plus 2. The derivative of that function is negative, so that means that that real value function decreases, and these a sub k's just take on values at the integer inputs, so that would show that the sequence decreases as well. In the end, because the sequence a sub k decreases to 0, the alternating series test applies, and it tells us that our alternating series converges. Unfortunately, our alternating series test doesn't tell us to what this series converges, but in a later screencast, we'll see how to use the alternating series estimation theorem to estimate the sum of this series as close as we like. Just as a reminder of why this alternating series converges, let's look at a few of the partial sums of this series. We have the first partial sum is just 6 fifths. The second partial sum is s1 minus the second term, 6 eighths, which is about 0.45. The third partial sum is the second partial sum plus the third term, 6 elevenths, that's about 0.995. And the fourth partial sum is the third partial sum minus the fourth term, so that's s3 minus 6 thirteenths, about 0.57. If we plot these partial sums, s1 through s4 on the number line, we can see that s1 is about 1.2, s2 is about 0.45, s3 is about 0.99, s4 is about 0.57. We see that the partial sums of our series oscillate around some point that is the actual sum of the series. Now, since the terms a sub k decreased to zero, that'll force the sequence of partial sums to actually collapse down onto the sum of the series. Now, a use of technology shows that the 1,000th partial sum is about 0.758, and the 10,000th partial sum is about 0.759. So we can be pretty confident that the series, sum negative 1 to the k plus 1, 6 over 3 k plus 2, has a sum that's pretty close to 0.76. Let's look at another example. Consider the series, sum k goes from 1 to infinity, 2k over k squared plus 2. Pause the video for a moment and determine if the alternating series test applies, and if it does, what conclusion can we draw? Resume the video when you're ready. If we write out the first few terms of this series, when k is 1 we have 2 thirds, when k is 2, 4 sixths, when k is 3, 6, 11ths, notice that all these terms are positive, so this series is not an alternating series, and we can't even apply the alternating series test. And if we want to determine the convergence or divergence of this series, we'll have to use one of our other tests that apply just a series of positive terms, like the comparison test, the ratio test, or the integral test. Let's do one more example. Consider the sum k equals 1 to infinity minus 1 to the k plus 1, k squared over k squared plus 3. Pause the video for a moment and determine if the alternating series test applies. If it does, what conclusion can you draw? Resume the video when you're ready. Let's look at the first few terms of this series. When k is 1, we get a quarter. When k is 2, we get negative 4 sevenths. When k is 3, we get 9, 12ths, and so on. So the terms in this series do alternate between positive and negative, and the alternating series test applies. Now the positive terms in this series have the form a sub k is k squared over k squared plus 3. But the limit of k squared over k squared plus 3 as k goes to infinity is 1, and so the sequence of positive terms in this series does not converge to zero. And the alternating series test says that the alternating series converges if and only if the sequence of kth terms decreases to zero, since the sequence of kth terms here doesn't even have a limit of zero, our alternating series diverges by the alternating series test. So let's summarize. The alternating series test says that the alternating series sum minus 1 to the k plus 1 a sub k converges if that sequence of the positive terms a sub k decreases to zero. The alternating series test also says that the alternating series diverges. If the sequence of positive terms a sub k does not decrease to zero. An important point, a critically important point to note is that the alternating series test applies only to alternating series, not to any other kind of series. And if a series is not an alternating series, then we need to apply one of our other tests to determine convergence or divergence. That concludes our screencast on using the alternating series test. Please come back soon.