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In this video, we will learn how to determine if a series is absolutely convergent, conditionally convergent, or divergent.
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We know that if a series is called convergent, it means that the partial sums approach a specific limit.
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But what does it mean for a series to be called absolutely convergent?
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We say a series is called absolutely convergent if the series of absolute values is convergent.
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One thing to notice is that if π π is a series with positive terms, then the absolute value of π π is equal to π π.
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So absolute convergence implies convergence.
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First, letβs look at an example where we have to determine whether a series is absolutely convergent.
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Is the series the sum from π equals one to β of negative one to the power of π add one over π squared absolutely convergent?
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Remember to test for absolute convergence, we need to check whether the series of absolute values is convergent, in other words, is the sum from π equals one to β of the absolute value of negative one to the power of π add one over π squared convergent.
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First of all, notice that negative one to the power of π add one is always going to be either one or negative one, depending on whether the power is even or odd.
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So if we take the absolute value of negative one raised to the power of π add one, this is always going to be one.
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We know the π runs from one to β.
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So π squared is always going to be positive.
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So we can actually rewrite this as one over π squared.
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Remember that weβre trying to determine whether this converges or diverges.
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But we actually recognize the sum from π equals one to β of one over π squared to be a series that we know.
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Itβs a π-series.
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So we use the fact that a π-series converges if π is greater than one and diverges if π is less than or equal to one.
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So for our question, we see that this is a π-series with π equal to two.
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Because this is greater than one, we can say that the sum from π equals one to β of one over π squared is convergent.
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So because we found the series of absolute values to be convergent, then the series is absolutely convergent.
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Interestingly, if we find a series, which is not absolutely convergent, it may still be convergent.
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We call this conditional convergence.
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A series is conditionally convergent if the series is convergent but not absolutely convergent.
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In other words, the sum from π equals one to β of the absolute value of π π diverges.
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But the sum from π equals one to β of π π converges.
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And if a series is not absolutely convergent and itβs not conditionally convergent, then itβs divergent.
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Letβs see an example of conditional convergence.
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Is the alternating harmonic series the sum from π equals one to β of negative one to the power of π add one multiplied by one over π absolutely convergent, conditionally convergent, or divergent?
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Letβs firstly remember that a series is absolutely convergent if the series of absolute values is convergent.
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And a series is conditionally convergent if the series of absolute values diverges but the series converges.
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And otherwise, the series is divergent.
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So letβs start by testing for absolute convergence.
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We can see that negative one raised to the power of π add one is always going to give us one when we take the absolute value.
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So this is in fact the same as the sum from π equals one to β of one over π.
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But this is actually a series that weβre familiar with.
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Itβs the harmonic series.
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And we know that the harmonic series diverges.
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So the alternating harmonic series is not absolutely convergent.
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But is it conditionally convergent or divergent?
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So our next step is to test the alternating harmonic series for convergence.
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Because thatβs an alternating series, we can do this with the alternating series test.
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Recall this says that for an alternating series, the sum of negative one to the power of π add one multiplied by π π if π π is decreasing and the limit as π approaches β of π π is equal to zero, then π π is convergent.
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So for the alternating harmonic series, we can say that π π equals one over π.
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So is π π decreasing?
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Well, as π increases, one over π does decrease.
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So that condition is satisfied.
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But does the limit as π approaches β of π π equal zero?
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Well, the limit as π approaches β of one over π is going to be one over β, which we know is zero.
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So that condition is satisfied.
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So because we found that the alternating harmonic series is not absolutely convergent, but it is convergent, we can conclude that the alternating harmonic series is conditionally convergent.
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We can summarize the check for absolute convergence, conditional convergence, and divergence in a helpful diagram.
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Letβs say we want to find out whether the series π π is absolutely convergent, conditionally convergent, or divergent.
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We begin by testing whether the series of absolute values is convergent or divergent.
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Letβs say that we find that the series of absolute values is convergent.
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Then, the series π π is absolutely convergent.
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But if we find that the series of absolute values is divergent, then the series π π is not absolutely convergent.
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But it may still be conditionally convergent.
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So we try a different test on the series π π to check for convergence, for example, the alternating series test.
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And if we find that the series π π converges, then we say that the series π π is conditionally convergent.
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But if we find that the series π π diverges, then we conclude that the series π π is divergent.
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So these are the three possible conclusions that we can draw.
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So letβs now see some more examples.
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Consider the series the sum from π equals one to β of sin of π over π cubed.
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Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
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Recall that a series π π is absolutely convergent if the series of absolute values is convergent.
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And if we find that the series is not absolutely convergent, it may still be conditionally convergent.
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So we then test the series for convergence or divergence.
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So letβs begin by testing this series for absolute convergence.
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So we want to find out whether the sum from π equals one to β of the absolute value of sin of π over π cubed is convergent or divergent.
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Well, because π only runs through positive values from one to β, π cubed is always going to be positive.
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So this is just the sum from π equals one to β of the absolute value of sin of π over π cubed.
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Now we know that sin of π will always be between negative one and one.
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So we can say that the absolute value of sin of π will always be less than or equal to one, which means that we can write the absolute value of sin of π over π cubed is less than or equal to one over π cubed.
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Writing it this way allows us to do a direct comparison.
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Recall that this means if π π is less than π π and the sum from π equals one to β of π π converges, then the sum from π equals one to β of π π also converges.
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And one over π cubed is actually a series we recognize.
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Recall that a π-series is a series of the form the sum for π equals one to β of one over π to the π power.
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And this converges if π is greater than one and diverges if π is less than or equal to one.
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So one over π cubed is a π-series with π equal to three.
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So one over π cubed converges.
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So by direct comparison, the absolute value of sin of π over π cubed also converges.
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Then, because we found the series of absolute values to be convergent, then our series the sum from π equals one to β of sin of π over π cubed is absolutely convergent.
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State whether the series the sum from π equals one to β of negative one to the power of π add one multiplied by two over the square root of π add one converges absolutely, conditionally, or not at all.
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Firstly, recall that for a series π π, this is absolutely convergent if the series of absolute values converges.
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And itβs conditionally convergent if the series of absolute values diverges.
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But the series itself converges.
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So letβs first of all find out whether this series is absolutely convergent or not.
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This means testing whether the series from π equals one to β of the absolute value of negative one to the power of π add one multiplied by two over the square root of π add one is convergent or divergent.
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Well, negative one to the power of π add one is always going to be one or negative one.
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But if we take the absolute value, it will always be one.
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Whereas two over the square root of π add one will always be positive because π runs through positive values.
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So we can write this as the sum from π equals one to β of two over the square root of π add one.
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Then, we can use the constant multiplication rule to bring the two to the front of the sum.
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From here, we need to work out whether this series converges or diverges.
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One way we can actually do this is with a direct comparison with the harmonic series.
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Because for π is greater than two, we have that one over π is less than one over the square root of π add one.
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And we know that if we have π π less than π π where π π diverges, then π π also diverges.
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And we know that the sum from π equals one to β of one over π is the harmonic series which diverges.
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Then, the sum from π equals one to β of one over the square root of π add one also diverges.
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So we found that the series of absolute values diverges, which means that this series is not absolutely convergent.
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But it could still be conditionally convergent.
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So weβre going to test the series itself for convergence.
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So letβs clear some space.
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We can firstly bring the constant two to the front of the sum.
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And then, if we look at this negative one to the power of π add one, this creates an alternating series because it makes the terms alternate between positive and negative.
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So we can decide whether this series is convergent or divergent using the alternating series test.
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Remember that this says for a series π π, where π π is equal to negative one to the power of π add one multiplied by π π, if π π is decreasing and the limit as π approaches β of π π is equal to zero, then the series π π is convergent.
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So for our series π π is equal to one over the square root of π add one.
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But is this decreasing?
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Well, for π π to be decreasing, we need π π to be greater than π π add one.
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Well, we can see that as π increases by one, the square root of π add one is going to get bigger.
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So one over the square root of π add one is going to decrease.
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So π π is decreasing.
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Then, we need to check whether the limit as π approaches β of π π is equal to zero.
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In other words, is the limit as π approaches β of one over the square root of π add one zero?
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Well, the square root of π add one is increasing as π gets bigger.
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So this will be one over β.
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So as π approaches β, then one over the square root of π add one approaches one over β.
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And so the limit as π approaches β is zero.
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So both of these conditions are satisfied.
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So this series is convergent.
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Remember that we said that a series is conditionally convergent if the series of absolute values diverges but the series converges.
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And thatβs exactly what weβve found here.
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The series of absolute values was divergent.
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But we found the series itself to be convergent.
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So we can conclude that this series converges conditionally.
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But why is that helpful to differentiate between absolutely convergent and conditionally convergent series?
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Well, absolutely convergent infinite series holds some of the same properties as finite sums.
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For example, if we have a finite sum, then any rearrangement of the terms still gives the same sum.
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And this also holds true for an absolutely convergent series.
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Any rearrangement yields the same sum.
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But this is not true for a conditionally convergent series.
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This is because rearranging the terms of a series changes the partial sums.
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So this can change the limit of the partial sums when some of the terms are negative.
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So we donβt have that issue with absolutely convergent series.
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But of course, this doesnβt apply to conditionally convergent series.
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As an example, the alternating harmonic series, which weβve seen is convergent, can be shown to converge to the natural log of two.
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But if the terms in the series are rearranged so that every positive term is followed by two negative terms, this does change the value of the sum.
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So with a conditionally convergent series, rearrangement changes the relative rate at which positive and negative terms are used and in turn changes the sum of the series.
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In fact, we can actually use this to rearrange a conditionally convergent series to converge to any value we want.
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But itβs beyond the scope of this video to go through these proofs in detail.
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Letβs now summarize some of the main points.
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A series π π is absolutely convergent if the series of absolute values is convergent.
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And a series π π is conditionally convergent if the series of absolute values is divergent, but the series itself is convergent.
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And finally, if π π is an absolutely convergent series with sum π , then any rearrangement of π π yields the same sum π .