11–27. Alternating Series Test Determine whether the following...

Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

Accepted Answer

Hello there. Today we're gonna 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 + 1 multiplied by 3 divided by 4 + 5 converge. Awesome. So it appears for this particular prom we're asked to take the series that is provided to us by the prom itself, and we're asked to determine whether or not it converges. So our final answer will be either a yes, the series converges, or B, no, the series diverges. So now that we know what we're ultimately trying to solve for, our first step that we need to take, or rather the first. The approach we should take with this particular prompt is we need to recall and use the alternating series test. So we need to suppose that we have a series such as the sum of a subscript N, and either a subscript N is equal to parentheses -1 to the power of N. Or a subscript N is equal to -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 N. So then, if our first condition, the limit as N approaches positive infinity of B subscript N is equal to 0. And 2 curly brackets of, so we're gonna have curly. Brackets of B subscript N. Of, so this is curly brackets of B sub group N is a decreasing sequence, so I put an arrow pointing down. That will mean that the series, the sum. Of a subscript N is convergent or it converges. So that will mean our first major step to solve this problem, keeping in mind these, keeping in mind that this particular approach that we're using is the alternating series test. So keeping in mind that we're once again using the alternating series test. Our first step is we need to identify B subscript M. So the series alternates, and we need to note that the series alternates due to parentheses -1 to the power of M + 1. So that will mean that B subscript M will be equal to M power of 3 divided by M 4 + 5. And for any value of m is greater than or equal to 1, both the numerator and the denominator will have a positive value. So that will mean that a positive divided by a positive will be greater than or equal to 0. Fantastic. So with that in mind, therefore, undoubtedly we could state that B subscript M will be equal to M 3 divided by m 4 + 5 is greater than or equal to 0 or all m is greater than or equal to 1. Fantastic. So that will mean that our second step is we need to check the limit condition. So we need that the limit as M approaches positive infinity of B subscript M is equal to 0. We now need to compute the limit, so the limit. As M approaches positive infinity of M 3 divided by M 4 + 5. Fantastic, is equal to the limit as M approaches positive infinity of 1 divided by M divided by 1 + 5 divided by M to the power of 4 is equal to 0, which means as a result, the condition is satisfied, so I'll put a check mark right next to it. So since the condition is satisfied, that will mean that our 3rd step is we need to check if B subscript M is decreasing. So that means we need to verify that B subscript M decreases with M, so that will mean that B subscript M is equal to M power of 3 divided by M 4 + 5 is equal to 1 divided by M multiplied by 1 divided by 1 + 5 divided by M to the power of 4. And we need to note that as M increases, 1 divided by M decreases, and also 1 + 5 divided by M to the power 4 increases towards 1. So therefore, as a result, that means that B subscript M decreases. For all M is greater than or equal to 1, so that means that the condition is satisfied. So I put a check mark to indicate that the condition is satisfied. So that means that our 4th and final step is we need to make our final conclusions. And since both the alternating series test conditions are satisfied, which once again, the first condition is the limit as M approaches positive infinity of B subscript M is equal to 0, and that our second condition that B subscript M is decreasing. That means as a result, the series. Which is the sum evaluated from M is equal to 1 to positive infinity of parentheses -1 to the power of M + 1 multiplied by 3 divided by M 4 + 5. Is converging or it converges and that's our final answer. So I'll just put C for converging for short. So C denotes converging, and that is our final answer. Hooray, we did it. So going back to look at our multiple choice answers, our final answer once again without a doubt will be multiple choice answer letter A. Which is, yes, the series converges. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

Key Concepts

Alternating Series Test

Limit of the Terms

Behavior of Rational Functions at Infinity

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