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 ∞) ((k / (k + 1)) × 2k²)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to use the root test to determine whether the series, the sum of N equals 1 to infinity of the quantity of N divided by the quantity of N + 2 in quantity, and that fraction is raised to the quantity of 3 N 2 in quantity converges absolutely or diverges. So, here we're asked to apply the root test to our series, and we call for the root test that if we have these series, which is going to be the sum of a of N, then we want to look at the limit as N approaches infinity of the nth root. Of the absolute value of a sub N. And if this limit is less than one, then our series is going to converge, absolutely. If this limit. is greater than one, then our series diverges. If this er this limit is equal to one, then this test is going to be inconclusive, so we'd have to use a different test in order to determine whether the series converges or diverges. So, for the series that we're looking at, we're going to want to look at the limit. As N approaches infinity. Of the quantity of the absolute value of the quantity of N divided by the quantity of N plus 2 in quantity, and that quantity is raised the quantity of 3 N squad. In quantity, and then that is going to be raised to the quantity of 1 divided by N in quantity. So here we've just rewritten the N3 as raising that quantity to the power of 1 divided by N. Now, we can simplify this expression, so this is equal to the limit. As in per infinity. Of the quantity of N divided by the quantity of N plus 2 in quantity, and that quantity is raised to the quantity of 3 N. In quantity. And so here we have removed the absolute value since um our end value is going to be a positive quantity. So the quantity of N divided by the quantity of N + 2 in quantity is always going to be positive, so we can drop the absolute value sign. Now, instead of trying to take this limit, we're actually going to want to look at taking the limit as it approaches infinity of the natural log of this quantity. So we're going to actually look at the limit. As in approaches infinity. Of the natural log of the quantity of N divided by the quantity of N + 2 in quantity, and that fraction is raised to the quantity of 3 N in quantity. Which used the properties of logarithms, this is going to be equal to the limit as N purchases infinity of 3N multiplied by the natural log of the quantity, and here rewrite our argument as 1 minus 2 divided by the quantity of N + 2. In quantity And by writing our natural law function in this matter, We can approximate the value of our natural log function. So the recall that the natural log of the quantity of 1 minus 2 divided by the quantity of N + 2 in quantity is going to be approximately equal to minus 2 divided by the quantity of N + 2 in quantity. So that means that our limit is going to be equal to the limit. As N approaches infinity of 3 N multiplied by the quantity of minus 2 divided by the quantity of N + 2. In quantity And so now we just need to simplify this expression. So in doing so, this is going to be equal to the limit. As in approaches infinity. Of minus 6 multiplied by the quantity of N divided by the quantity of N plus 2n quantity, dividing both the numerator and denominator of a fraction by N, we see that this is going to be equal to the limit as N approaches infinity. Of minus 6 multiplied by the quantity of 1 divided by the quantity of 1 plus 2 divided by N in quantity. And as N approaches infinity, the quantity of 2 divided by N approaches 0. So applying the limit, this is going to be equal to minus 6 multiplied by the quantity of 1, divided by the quantity of 1 plus 0, and quantity which is equal to minus 6. So, if we look at the original limit that we're looking for, which is going to be the limit. As in approaches infinity. Of the quantity of N divided by The quantity Of N + 2 in quantity, that quantity is raised to the quantity of 3 N in quantity, that this is going to be equal to E raised to the minus 6 power. And this value is actually less than one. That means that our series actually converges absolutely. So, that is the answer that we were looking for for this problem. 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 ∞) ((k / (k + 1)) × 2k²)

Key Concepts

Ratio Test

Root Test

Absolute Convergence

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