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 = 1 to ∞) (1 + 2 / k)ᵏ

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says consider the series the sum of M equal to 1 to infinity of the quantity of 1 plus 1 divided by M and quantity rates to the power M. Does this series converge or diverge? And we're given two possible choices as our answers. For choice A, we have converge, and for choice B, we have diverge. Now to test whether the series converges or diverges, we're going to be using the divergence test. So we call for the divergence test that if the limit as M approaches infinity of a sub M is not equal to 0 or does not exist, that means that that series will diverge. So, first thing we need to do is identify a sub M, and based on the series that we're given, AM is going to be equal to the quantity of 1 plus 1 divided by M in quantity raised to the power M. So that means that we need to calculate the limit as M approaches infinity of a of M, which is equal to the limit. As M approaches infinity of the quantity of 1 plus 1 divided by M and quantity raised to the power M. Now, if we try to take this limit, we're not gonna be able to do so, since we have M as an exponent. So what we're going to want to do is take the natural log of both sides. So that means that the limit. As M approaches infinity of the natural log of a subm is going to be equal to the limit as M approaches infinity of M multiplied by the natural log of the quantity of 1 plus 1 divided by m in quantity. And so here we've used the property's logarithms to move the exponent into a coefficient out in front of the natural log function. Now if we try to Evaluate this limit by direct substitution. We'll see that we'll get infinity multiplied by 0, which is an indeterminant form, but it's also not a form that we can use Lopital's rule for. So we want to do is rewrite our expression here as being equal to the limit as M approaches infinity of the natural log of the quantity of 1 plus 1 divided by m in quantity divided by the quantity of 1 divided by m in quantity. So here if we try to take the limit as M approaches infinity, we will get 0 divided by 0, which again is an indeterminate form, but one that we can apply Lopital's rule for. So applying Lopital's rule, this is going to be equal to the limit as M approaches infinity, and we call for Lopital's rule that we need to take the derivative, both the numerator and denominator, with respect to M. So taking the derivative of the numerator, we will get the quantity of 1 divided by the quantity of 1 plus 1 divided by m in quantity, and then that gets multiplied by the quantity of minus 1 divided by m squared in quantity, and that is divided by when we take the derivative of denominator, we get the quantity of minus 1 divided by m squared and quantity. So simplifying this expression, this is going to be equal to the limit, as M approaches infinity of 1 divided by the quantity of 1 plus 1 divided by M in quantity. And as M approaches infinity, the quantity of 1 divided by M approaches 0. So applying the limit, this is going to be equal to 1 divided by the quantity of 1 plus 0. Which, when we evaluate this expression, it's just going to be equal to one. However, we're not interested in the limit as M approaches infinity of the natural log of A of M. We're interested in the limit. As M approaches infinity of ace of m, so we need to exponentiate both sides of that expression. So that means that the limit as M approaches infinity of a of M is going to be equal toe raised to the first power, which is just E. Now this here is not equal to 0, so that means that our series actually diverges. So that is the answer that we're looking for for this problem. And when we compare our answer here to 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 = 1 to ∞) (1 + 2 / k)ᵏ

Key Concepts

Convergence of Infinite Series

Root Test

Limit Comparison and Growth of Terms

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