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 = 2 to ∞) (5lnk) / k

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says that the series, the sum of M equal to 5 to infinity of 6 multiplied by the natural local of M divided by M, converge, or diverge. And we're given two possible choices as our answers. For choice A, we have the series converges, and for choice B, we have the series diverges. So, in order to determine whether our series here converges or diverges, we're going to use the integral test. So we call for the integral test that we need to calculate the integral from N to infinity of F of X. DX And if this integral is finite or converges, then so does our series. And if this integral diverges, meaning it approaches plus or minus infinity, then so does our series. Now, looking at our series, we see that F of X. is going to be equal to 6 multiplied by the natural log of X. Divided by X, and N is going to be equal to 5, since the series summation starts at M equals to 5. So that means our integral is going to be equal to the integral. From 5 to infinity. Of 6 multiplied by the natural log of X, divided by X. DX. So now we just need to evaluate this improper integral. So, the first thing we're going to do is make a substitution. So we're gonna set U equal to the natural log of X, and that means that DU is going to be equal to 1 divided by XD X. So rewriting our integral using the substitution. This is going to be equal to the integral, and here we have to change our limits of integration since we're changing our variables. So when X is equal to 5 for the lower limit of integration, U is going to be equal to the natural log of 5. And for the upper limit of integration, as X approaches infinity, you is going to be approaching the natural log of affinity, which is still infinity. And for the integrand, we will have 6 multiplied by U D U. So now we need to evaluate this improper integral. So we're going to rewrite this as being equal to the limit, as T approaches infinity. Of the integral from the natural log of 5 to T of 6UDU. And so now we can perform the integration. So this is going to be equal to the limit. As T approaches infinity. Of 6 multiplied by, and here we integrate you with respect to you, we get The quantity of U squared divided by 2 in quantity, and this needs to be evaluated from the natural log of 52 T. So substituting in our limits of integration, this is going to be equal to the limit as T approaches infinity of, and here we can simplify the coefficient. So we'll have 6 divided by 2, which is 3. This is going to be multiplied by the quantity. Of T squared. Minus The quantity of the natural log of 5 in quantity squared in quantity. Now, as T approaches infinity, the quantity of T2d also approaches infinity. And we're gonna be subtracting alpha consonants, so that really doesn't change the results. So that means that this integral is going to be equal to infinity. So, that means that our integral here actually diverges. And since our integral here diverges, that means that our series also diverges. So that is the answer that we're looking for for this problem. And so if we look at our answer choices, we see that 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 = 2 to ∞) (5lnk) / k

Key Concepts

Convergence of Infinite Series

Comparison and Limit Comparison Tests

Integral Test

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