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 ∞)cos(1 / k⁹)

Accepted Answer

Determine whether the series, the sum from 1 to infinity of tangent, 2 divided by the 5th converges, and we have two possible answers being convergence or divergence. Now, for this, we'll make use of the limit comparison tests. We are using this test because we have a positive term series. And we also have 2 divided by into the 5th, which is also a known P series. So, for the limit comparison tests, we have L equals the limit. As M approaches affinity of two series that are positive, a m divided by BM. If we have L and it exists. And 0 is less than 1, which is less than infinity. Either the sun Of AM And the sum Of BM converge or diverge. This means they will both converge or both diverge. Let's write our general term. This is a sub N equals the tangent. Of 2 divided by N to the 5th. Now we know this is a positive term series. And for a large N, 2 divided by N to the 5th is small. We will then compare this to B sub in. Which is 1 divided by n to the 5th. Now, we can go ahead and write this by taking a substitution. Let's say X equals 2, divided by N to the 5th. We know X is greater than 0. And it goes to 0. As in approaches infinity. So we can rewrite this as the tangent. Of X divided by. X divided by 2. Which is this to Tangent X divided by X. Now we can use our limit. We have the limit. As x approaches 0. Of Tangent X divided by X multiplied by 2. Now, we know the limit, as tangent X divided by X approach to 0 is 1, which means we have the limit of 2 multiplied by 1, which is just 2. We can then confirm that our limit. As in approaches infinity. Of a subn divided by bn. S equals the 2 Now we also then know That the sum of 1 divided by into the fifth converges by the P series. This implies then that the limit of our series also converges, which means our answer is answer A, converges. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)cos(1 / k⁹)

Key Concepts

Convergence of Infinite Series

Limit Comparison and Behavior of Terms

Convergence Tests for Series

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