42–76. Convergence or divergence Use a convergence test of you...

Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k⁵ e⁻ᵏ

Accepted Answer

Welcome back everyone. In this problem, we want to figure out if the series for MQbed multiplied by 3 negative M between M equals 1 and infinity converges or diverges. A says it converges and B says it diverges. Now how can we know which is which? Well, we can use the raha test. Recall that by the re of test, OK. By the ratio test for a series AM with a greater than 0, when the limit L is equal to the limit as M approaches infinity of AM + 1 divided by Am, in other words, each term and their consecutive term, the ratio of the term and its consecutive term. Then If L is less than 1, the series converges. Absolutely, OK. If L is greater than 1, the series diverges. OK. Let me write that properly here, dive verges. And if L equals 1, the test is inconclusive. So if we can set AM and find the ratio of AM + 1 to AM, then we can solve for L and figure out what our outcome will be. So let's let, let me put this in red here, let's let AM be equal to M cubed multiplied by 3 to the negative M. Then if we apply the ratio test. Then I limit As M approaches infinity, is going to be the absolute value, OK, of M + 1, OK. To sorry cubed multiplied by 3 to the negative M + 1 because now remember we're replacing everything with. M with M + 1, OK? And all of that is going to be divided by AM, which is M cubed multiplied by 3 to the negative M. So let me just, uh, let me just explicitly state that here just for clarity. So this is our limit as M approaches infinity of AM plus 1 divided by AM, OK. Now if we think about it here. If we take what's on our right hand side, OK, then we could cancel our 3 to the negative M uh in our numerator and denominator, and that is gonna leave us with the limit as M approaches infinity of M + 1 cubed divided by M cubed multiplied by 3-1. Or 1/3. Now, if we simplify that even further, notice that. In this expression, OK, or, or, or quotient, we could divide through by M and then rewrite it, OK, as 1 + 1 divided by M, OK, or Q. And no, in this case we'll have this expression multiplying a third, and if we find the limit as M approaches infinity, then all of this will become one, OK, because 1 divided by a very large number will be very close to 0. So essentially it will be 1 and 1 cubed is 1. So this is gonna be 1 multiplied by 1/3, which will be equal to 1/3. Now remember in our test we said that if if L is less than 1, the series converges absolutely. Here it is less than 1, so that means that our series converges. By the racial test. Therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k⁵ e⁻ᵏ

Key Concepts

Infinite Series and Convergence

Comparison and Limit Comparison Tests

Behavior of Exponential vs. Polynomial Functions

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