11–86. Applying convergence tests Determine whether the follow...

Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Does the series, the sum evaluated from M equals 1 to positive infinity of parentheses -1 to the power of M. Multiplied by M divided by m to the power 2 plus 2 m converge. Awesome. So we're asked to determine for this particular problem. Does this particular series converge? Will it be a yes, the series converges, or B no, the series diverges. So now that we know what we're ultimately trying to solve for. What we must first do is we need to recall and use the alternating series test. And as we should recall, using the alternating series test, we need to suppose that we have a series which we will say that this series is the sum of a subscript N, and either a subscript N is equal to parentheses -1 to the power of N multiplied by B subscript N. Or a subscript N will be equal to parentheses -1 to the power of N + 1 multiplied by B subscript N. Where B subscript N is greater than or equal to 0. For all values of N. Then if the limit as N approaches positive infinity of B subscript N will be equal to 0, and curly brackets of B subscript N will be a decreasing sequence, which I'll call DS for short. That will mean that as a result, that the series, the sum of a subscript N will be convergent, which I'll just call simply C. So with that in mind, our first step is we need to identify the value of B subscript M. And in order to do that, we need to write the series in the form of -1 parentheses to the power of M multiplied by B subscript M. Where B subscript M is greater than or equal to 0. So you go ahead and write that B subscript M is equal to M divided by M 2 + 2M. Which is equal to M divided by M multiplied by parenthesis M + 2. And what we did is we just factored out the denominator. And this will be equal to when we simplify, 1 divided by M plus 2. So, so we need to note that B subscript M is greater than 0, for all M is greater than or equal to 1. Fantastic. Our second step now is we need to check if B subscript M decreases. So with that in mind, we need to compare B subscript M and B subscript M + 1. So, with that in mind, we need to go ahead and write that B subscript M is equal to 1 divided by M + 2. And B subscript M + 1 is equal to 1 divided by M + 3. And since 1 divided by M + 3 is less than 1 divided by M + 2, The sequence B subscript M is decreasing, which I'll simply call D. For decreasing. So now, with that in mind, our third step is we need to check the limit. So we need to take the limit as M approaches positive infinity, so the limit as M approaches positive infinity of B subscript M, which will be equal to the limit, as M approaches positive infinity of 1 divided by M + 2 will be equal to 0. Fantastic. Our 4th step is we need to recall and apply the alternating series tests. And as we should recall, all the conditions are satisfied, meaning our first condition. Is that B subscript M is greater than or equal to 0. Our second condition is that B subscript M decreases, so that's two conditions met. Our third condition that we need to meet is met because the limit as M approaches positive infinity of B subscript M is equal to 0. So therefore, by the alternating series tests the series, which is the sum that is evaluated from M equals 1 to positive infinity of parentheses -1 to the power of M multiplied by M divided by M 2 + 2 M. Does indeed converge, which I'll call C. So that means that our final answer, without a doubt when we look at our multiple choice answers, will have to be multiple choice answer letter A, which once again is yes, the series converges, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)

Key Concepts

Alternating Series Test

Limit of the General Term

Comparison Test

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