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) / (k!)

Accepted Answer

Welcome back everyone. Use the ratio test to determine whether the series sigma from J equals 1 to infinity of 5 power of J divided by J factorial converges absolutely or diverges. For this problem, let'st recall that the ratio test requires us to find the limit l where L is equal to the limit as J approaches infinity of the absolute value of the ratio between a subscript J + 1 and a subscript J. So let's begin by finding each term. We're going to write limit as J approaches infinity of the absolute value of. Now A subscript J + 1 is going to be 5, raises the power of J + 1 because now J becomes J + 1, right? divided by J + 1 factorial. And we're dividing by a subscript J. We're basically multiplying by the reciprocal of 5 power of J divided by J factorial. So we're multiplying by J factorial. Divided by 5 to the power of J. And now we're going to try and simplify this expression. So we have limit as J approaches infinity of the absolute value of. 5 Multiplied by 5 to the power of G we're using the properties of exponents. This is divided by 5 power of J. We can write the same terms under one fraction. Multiplied by G factorial, which is divided by. G + 1 factorial can be written as G +1, multiplied by G factorial. And I notice that we can cancel out 5 to the power of J, we can cancel out G factorial, and we're going to end up with Limit as J approaches infinity. Of the absolute value of 5. Divided by J + 1. Now, we can drop the absolute value because J begins with 1, and the minimum value 1 + 1 is going to be positive. That's 2, right? So we can state that this is equal to 5. We can also factor out the constant. Multiplied by a limit as J approaches infinity of 1 divided by J plus 1. Now, as J approaches infinity, we can notice that our denominator is going to approach infinity, right? Because we're basically adding 1 to infinity and that's infinity, and 1 divided by an infinitely large number is going to give us 0. So we can say that our limit is 5 multiplied by 0, which is 0. Now, based on this outcome, let's recall that the ratio test says that if L is less than 1, then our series converges absolutely. So, for this problem, we have our final answer. We have our limit, which is 0, and that's less than 1. And according to the ratio test, whenever our limit is, less than 1 hour series converges absolutely. Thank you for watching.

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) / (k!)

Key Concepts

Ratio Test

Root Test

Absolute Convergence

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