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 ∞) ((-1)ᵏ⁺¹ × k²ᵏ) / (k! × k!)

Accepted Answer

Hello. In this video, we are going to be determining whether the given series converges absolutely or diverges by using the ratio test. Now the series that is given to us is an infinite series starting at J is equal to 1. Of the following series, -1 raised the power of J, multiplied by J raised the power of 5J, divided by J-factorial, raised the power of 5. Now, in order to use the ratio test, the ratio test states the following. We want to take the limit as J approaches infinity. Of the absolute value of AM plus 1 divided by AM. This is where A sub-M represents the original series that is given to us. And so what this means is that in order to use the ratio test, we want to go ahead and plug in the value M + 1, or in this case here, J + 1 into the original series, then we want to divide that by the original series itself. So, let's go ahead and set up the given limit. So we are taking the limit as J approaches infinity of the absolute value of J +1 plugged into our series. That is going to give us J +1, raise to the power of 5 multiplied by J + 1. Divided by J +1 factorial, raised the power of 5. Then we are going to multiple divide by the original series, but that is going to give us a complex fraction. So let's instead multiply by the inverse of the original series. That is going to be J factorial, race the power of 5, divided by J raise the power of 5J. So, there's a bit of simplification that we can do here. In the first term, if we were to distribute the exponent 5 into J plus 1, we can rewrite that exponent to be 5J plus 5. And in terms of factorials, we can go ahead and simplify the factorials by expanding the factorial term in the denominator. See, if we were to expand the J+1 factorial race to the 5th, we will be left with the following. We will have J+1 raise to the 5th power multiplied by J factorial raise to the 5th. And by expanding this factorial, we can eliminate the J factorials to the 5th power in both the numerator and denominator. So, that is going to allow us to simplify this to be the limit as J approaches infinity of the absolute value, J + 1. Raise to the 5 J plus 5, divided by J +1, raise the 5th power multiplied by J raise the 5 J. Now, there is one more simplification that we can do here. In the numerator, we can rewrite the exponent or the exponential term to be J +1, raise the 5 J power multiplied by J +1, raise the 5th power. And by rewriting the series this way, we can eliminate the J + 1 race the fifth power terms. So that is going to leave us with the limit as J approaches infinity of the quantity J +1 raised the 5J power, divided by J raises the 5J power. And because both the numerator and denominator are raised to the same exponential value, that means we can further rewrite the limit to be the limit as J approaches infinity of the quantity J plus 1 divided by J raised to the 5 J power. And further simplifying this is going to leave us with the limit, as J approaches infinity of 1 + 1 divided by J, raise the power of 5G. Now in this case here, if we were to further simplify, we will be left with ease of the power of 5, and this value itself is greater than 1 because the value of the limit is greater than 1 by the conditions of the ratio test, our series is going to be divergent, and that is going to be the solution to this problem. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all 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 ∞) ((-1)ᵏ⁺¹ × k²ᵏ) / (k! × k!)

Key Concepts

Ratio Test

Root Test

Absolute Convergence

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