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 ∞) (4k)! / (k!)⁴

Accepted Answer

Does the series sum from M equals 1 to infinity of 5m factorial divided by m factorial to the 5th converge or diverge? There are two possible answers being converges or diverges. Now, to solve this, we will first make use of the ratio test. This is because this series has positive terms that involve factorial or exponential light growth. So we can say the sum. Of a With Am being greater than 0. Now, to use ratio test, we will take the limit L equals the limit, as M approaches infinity of A M + 1 divided by AM. Now, if L is less than 1, We have absolute convergence. If we have L is greater than 1, we have divergence. If we have L equals 1, the test is inconclusive. So let's go ahead and write our ratio next. We have A M + 1 divided by A of M. What an Ecu With 5 multiplied by M plus 1 factorial. Divided by M + 1 factorial, raised to the 5th. All divided by 5m factorial divided by m factorial. To the 5th. We can rewrite this by taking our reciprocal. We get 5M + 5 factorial. Divided by by m factorial. Multiplied By M factorial to the 5th divided by M + 1 vectorial. To the 5th Now we can simplify each of our different parts. I'll put our first part in red here, which simplifies to 5M + 5, multiplied by 5M + 4, and so on until we get to 5M + 1. We then have our other products. This simplifies to 1 divided by M + 1. To the 5th Let's go ahead and rewrite our product now. We have AM plus 1 divided by AM is equals to the products. From k equals 1 to 5. A 5 musk. Divided by M + 1. Now we can find the limit of this, you know, the limit. As M goes to infinity. We will then write 5M + K. Divided by M + 1. This gives us the limit. As M goes to infinity. Of 5 plus k divided by M. Divided by 1 Plus 1 divided by m. This limit is just 5. Now we can find our L L then. Will be given by the limit As M goes to infinity of AM plus 1 divided by AM. Which in our case is just 5 multiplied by itself 5 times or 55. This gives us 3125. Now since It's greater than the 1, because it's 3125. Our series by the ratio test. Divergence. This is our final answer. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴

Key Concepts

Ratio Test

Factorials and Combinatorial Expressions

Convergence of Series

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