40–62. Choose your test Use the test of your choice to determi...

Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 3 to ∞) 1 / lnk

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says, does the series, the sum of M equal to 4 to infinity of 1 divided by the natural log squared of M converge or diverge. Now we're given two possible choices as our answers. For choice A, we have converges, and for choice B, we have diverges. Now, if we look at our series here, we see that it has a form of the sum of a sub M, where a sub M is equal to 1 divided by the natural log squared of M. Now, if you look at A M, we noticed that each term of M is going to be a positive term, since AM is going to be greater than 0 or M greater than or equal to 4. That means that we can either use the comparison test or the integral test. So in this case, we're going to use the integral test. So, we're going to want to look at the interval from N to infinity. Of F of X. DX. And if this interval is finite, then that means that our series is going to converge. However, if this integral approaches infinity, that means that our series is going to diverge. Now, in our case, F of X is going to be equal to 1 divided by the natural log squared of X. And we're gonna have 4 N, it's going to be equal to 4, since our series starts at M equal to 4. So that means that our integral is going to be equal to the integral from 4 to infinity. Of 1 divided by the natural log squared of M. DX. So now we just need to perform this integration. The first step is to make a substitution. So, we're gonna set U equal to the natural log of X. And that means that X is going to be equal to E rates to the U. And or DU, that's going to be equal to 1 divided by X. DX And that also means that DX is going to be equal to E U D U. So with these substitutions, We can rewrite our integral as being equal to the integral, and here we have to change our limits of integration since we're changing variables. So for the lower limit of integration, when X is equal to 4, U is going to be equal to the natural log of 4. And for the upper limit integration, as X approaches infinity. U, which is equal to the natural log of X, is also going to approach infinity, so we'll still have infinity as our upper limit of integration. Then for the integram, we will have E to the power U. Divided by U 2 DU. Now, if we look at our integrand here, we notice that for large values of U, that Eu is going to grow faster than U squad. So that means as you approaches infinity, our integrand here is going to approach infinity. And since our integran here is approaching infinity as us approaches infinity, that means when we perform the integration, it's also going to approach infinity. That means that this integral is just going to be equal to infinity. And since our integral here is approaching infinity, that means that our series is going to be diverging. So our series actually diverges. So that is the answer that we're looking for for this problem. And if you look at our answer choices, we see the correct answer choice corresponds to answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / lnk

Key Concepts

Convergence of Infinite Series

Comparison Test and Limit Comparison Test

Integral Test

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