11–86. Applying convergence tests Determine whether the follow...

Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2⁹k / kᵏ

Accepted Answer

Welcome back, everyone. Does the series sigma from M equals 1 to infinity of 34 M divided by M to the power of M converge? A says yes and B says no. For this problem, let's define our term A subscript M, which is 3 to the power of 4 M divided by M to the power of M. We will notice that both the numerator and the denominator contain an exponent in terms of M, right? Our numerator has 4 m and our denominator has M for the exponent. And whenever this is the case, we can apply the root test. It says that we want to evaluate the limit as M approaches infinity of the m root of a subscript M, and this is what we're going to do. We're going to take a limit as M approaches infinity of the m root of 3 raises to the power of 4, M divided by M to the power of M. Now, let's simplify, this is limits. As M approaches infinity of the m root of 3 raises the power of 4 M divided by the m root of M raises the power of M. Now, what we can do is simply consider our term we erase the power of 4M, and according to the properties of exponents, this can be written as we raise the power of 4, raise the power of M, which is equal to 81, raise the power of M, right? So that we can rewrite our limit as limit as M approaches infinity of the M through of 81 raises the power of M divided by the m root. Of m raises the power of M, which is equal to the limit as M approaches infinity of 81 divided by m. And now as M approaches infinity because we're dividing a number by an infinitely large number, we're going to get a limit of 0. According to the root test, if L is less than 1, and this is the case here, the series converges. If L is greater than 1, the series diverges, and if L is equal to 1, the test is inconclusive. According to the outcome. That L is less than 1 in this case, we can conclude that the series converges. So the answer to this problem is a. Yes, the series converges. Thank you for watching.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ

Key Concepts

Infinite Series and Convergence

Root Test for Convergence

Exponential and Factorial Growth Rates

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