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 ∞)(7ᵏ + 11ᵏ) / 11ᵏ

Accepted Answer

Welcome back, everyone. Does the series sigma from M equals 1 to infinity of 2 of M + 5 M divided by 5 power of M converge or diverge. A says converges and B says diverges. For this problem, let's define our term A subscript M, which is 2 of M plus 5 power of M divided by 5 power of M. We're going to apply the series divergence test, which says that we want to take limit as M approaches infinity of a subscript M. So we take a limit as M approaches infinity of 2, raises the power of M + 5, raise the power of M, divided by 5, raise the power of M. And we can evaluate this limit by performing division. We get limit as M approaches infinity of 2, raise the power of M divided by 5, raise the power of M. La 5 raise the power of M divided by 5 raise the power of M. Now using the properties of exponents, we can write our limit as limit as M approaches infinity of. First of all, we can include 1 because 5 power of M divided by 5 power of M is 1, and then we're adding 2/5, raise the power of M. Now because our fraction is between 0 and 1, not inclusive, as M approaches infinity, our fraction to the power of 5 raises the power of M is going to approach 0, right, because it has a value between 0 and 1 not inclusive. So when we raise such a fraction to an infinitely large number. We're going to get a 0 for the limit. Therefore, our limit is going to be 1 + 0, which is equal to 1. According to the series divergence test, if our limit is not equal to 0 or the limit does not exist, then the series diverges, which is exactly what we have. We can conclude that the series diverges. Well then, so the answer to this problem is be divergence. Thank you for watching.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(7ᵏ + 11ᵏ) / 11ᵏ

Key Concepts

Infinite Series and Convergence

Geometric Series

Convergence Tests (Comparison and Limit Tests)

Watch next