9–30. The Ratio and Root Tests Use the Ratio Test or the Root ...

Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says to use the ratio test to determine whether the series, which is the sum of J equal to 1 to infinity, of 4 rates to the power J divided by J rates to the power of 120, converges absolutely or diverges. So this problem wants us to use the ratio test. So recall for the ratio test that if we have the series, the sum of a sub J, that we want to look at the limits. As J approaches infinity of the absolute value of the quantity of a J plus 1 divided by a J in quantity, and if this limit is less than 1, then that means that our series converges absolutely. If this limit. is greater than one that or it approaches infinity, that means that this series diverges. And if this limit is equal to one, That means that this test is inconclusive. So we would have to use a different test to determine whether this series converges or not. So that means we're going to be looking at the limit. As J approaches infinity. Of the absolute value. Of the quantity of the quantity of 4, raised to the quantity of J plus 1 in quantity, divided by the quantity of J plus 1 in quantity raised to the power of 120 in quantity, and that's going to be divided by the quantity of 4 raised to the power J, divided by. J raised to the power of 120 in quantity. So we can actually simplify this expression. So this is going to be equal to the limit as J approaches infinity of the absolute value of the quantity of 4 multiplied by the quantity of J divided by the quantity of J plus 1 in quantity, and then that fraction is raised to the power of 120 in quantity. So, in our fraction there, we will want to divide both the numerator and denominator by J. So that means that this is going to be equal to the limit. As J approaches infinity of the absolute value of the quantity of 4 multiplied by the quantity of 1 divided by the quantity of 1 plus 1 divided by j in quantity and that fraction is raised to the power of 120, then in quantity. Now, as J approaches infinity, the quantity of 1 divided by J approaches 0. So evaluating this limit, this is going to be equal to the absolute value of 4 multiplied by the quantity of 1 divided by the quantity of 1 + 0 in quantity, and then that fraction is raised to the power of 120 in quantity, and evaluating this expression, we see that this is going to be equal to 4. And the value of 4 here for this limit is actually greater than 1. So since 4 is greater than 1, that means that this series diverges. So that is the answer that we're looking for, for this problem, that the series diverges by the ratio test. So, I hope that this explanation has been useful, and I'll see you in the next video.

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)

Key Concepts

Ratio Test

Root Test

Absolute Convergence

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