Determining Convergence or Divergence
Which of the serie...

Question

Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) (1 + 1/n)ⁿ

Accepted Answer

Welcome back everyone. Determine whether the series stigma from n equals 1 to infinity of 1 + 2 divided by n to the power of n converges or diverges. For this problem we're going to use the nth term test for divergence. Specifically, what we want to do is evaluate the limit as n approaches infinity of the nth term, and that's 1 + 2 divided by n to the power of n. In order to evaluate this limit, we are going to use the limit as k approaches infinity of 1 plus. 1 divided by k to the power of k and this limit is equal toe, right, so we're going to introduce a substitution. Let's suppose that 2 divided by n is equal to 1 divided by k, right. Then we can solve for n. n is going to be equal to 2 k multiplied by. I'm sorry, 2 k divided by 1, right, so that would be 2 k, meaning we can rewrite our limit as limits. If n approaches infinity, k approaches infinity as well because n is equal to 2k. So we get limit as k approaches infinity of 1. Plus 1 divided by k raises the power of 2 k. And we can now use the properties of exponents. We can just say raise the power of k. Squared Using the properties of limits, we can now express it as a limit. Ask approaches infinity of 1 + 1 divided by k to the power of k 2, which is equal to e2. Well done, and now according to the test for divergence, we basically have to compare it to 0 because it is not equal to 0, the series diverges. And that's our final answer for this problem. Thank you for watching.

Determining Convergence or DivergenceWhich of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)∑ (from n=1 to ∞) (1 + 1/n)ⁿ

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