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 ∞)sin(1 / k⁹)

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 is equal to 1 to positive infinity of sine of 1 divided by M to the power of 5 converge. Awesome. So it appears for this particular problem we're given a series and we're asked at this particular series either a, yes, the series converges, or B, no, the series diverges. So we're ultimately trying to determine whether or not the series converges or diverges. So with that in mind, Our first step that we have to take is we need to be able to look at our series and we need to be able to recognize and recall that we have a positive term series and because we have a positive term series, we can recall and use the limit comparison test with a known P series will be a convenient way to solve this particular problem. So with that in mind, for a limit comparison test, as we should recall, for positive sequences A subscript M and B subscript M, We can note that we can perform a limit comparison test if lowercase l, which I'll just call L, is equal to the limit, as m approaches positive infinity of a subscript M divided by B subscript M, and this exists. If 0 is less than L and L is less than positive infinity, then that will mean that the sum of A subscript M and the sum of B subscript M either both converge or diverge. So with that in mind, we can now begin our step by step solution. And our first major step we need to take to solve this problem is we need to write the general term. So the general term states that a subscript M will be equal to sign of 1 divided by M to the power of 5. So at this point we need to recall a note that for large values of M, the argument 1 divided by N to the power of 5 is positive and small, so that will mean, and let's make a note of that note of this, that will mean that sign of 1 divided by m the power of 5 is greater than 0. Therefore, as a result, we may compare with a positive sequence. So that will mean that we can go ahead and write that B subscript M is equal to 1 divided by M to the power of 5. Fantastic. Our second step now is we need to compute the limit. So we need to take the limit as M approaches positive infinity of a subscript M divided by B subscript M. Which will be equal to the limit. As M approaches positive infinity of, of 1 divided by M to the power of 5, divided by 1 divided by M to the power of 5. Fantastic. And at this point we need to take a moment here to pause and recall and use the standard small angle limit which, as we should recall for the standard small angle limit we can write that the limit as x approaches 0 of sign. Of X divided by X is equal to 1. So let us state that for this particular problem, X is equal to 1 divided by M to the power of 5. So that will mean that this will approach 0 as M approaches positive infinity. So therefore, our limit will equal 1, so we can safely go ahead and write that the limit. As M approaches infinity of sine of 1 divided by M to the power of 5, so it's gonna be 1 divided by M to the power of 5, divided by 1 divided by M to the power of 5. Will be simply equal to one. Fantastic. So our 3rd and final step now is we need to make our final conclusions. So since we know that the limit, lowercase l is equal to 1 and is finite and positive, And since the sum of B subscript M is equal to the sum of 1 divided by M to the power of 5 is a convergent P series, which I call C for convergent and PSSP series. So that will mean that P is equal to 5 and 5 is greater than 1. That means as a result, the limit comparison test implies that the sum of a subscript M also converges, so I'll put C in quotation marks with a check mark. So because this these Once again, since we used our limit comparison test, and it also implies that the sum of a subscript M converges, and we determine that the sum of B subscript M also converges, that means that our final answer without a doubt will have to be multiple choice answer letter A, which once again is yes, this series converges. So 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 ∞)sin(1 / k⁹)

Key Concepts

Convergence of Infinite Series

Comparison Test for Series

Behavior of sin(x) for Small x

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