Use the ratio test to determine whether ∑n=30^∞(6^n)/((2n)!)...

Use the ratio test to determine whether $\sum_{n=30}^{\infty} \frac{6^n}{(2n)!}$ converges or diverges. (a) Find the ratio of successive terms. Write your answer as a fully simplified fraction. For $n \ge 30$, $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{3}{\square}$ (b) Evaluate the limit in the previous part. Enter $\infty$ as infinity and $-\infty$ as -infinity. If the limit does not exist, enter DNE. $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0$ (c) By the ratio test, does the series converge, diverge, or is the test inconclusive? Converges

Transcript

00:03 All right, we are asked to use the ratio test to check for the convergence of this series.

00:10 The series is the sum from n equaling 11 to infinity, 8 to the n over, and then 2n factorial, quantity 2n factorial.

00:18 All right, well, our ratio test says we can check the limit as n approaches infinity of, and it's the absolute value of a sub n plus 1 over a sub n.

00:31 So what that means to do is literally take out your original term, and anywhere you see an n, replace it with an n plus 1.

00:38 So it's 8 to the n plus 1 divided by, and it is two times instead of n, it's n plus 1 factorial.

00:47 All right.

00:48 And now i'm going to divide that by the original.

00:51 All right.

00:53 So let's just extend these guys down.

00:56 And typically i don't write it out this way, but i think this is what it was actually looking for, something like this.

01:02 To n factorial.

01:04 So there's my division by the original.

01:06 I don't typically write it this way because when i divide by this fraction, i just change it right away to be multiplication by the reciprocal.

01:14 So what we have to try and evaluate here is the limit as n approaches infinity.

01:19 I can get rid of the pesky absolute value symbols because none of this can be negative if you think about all these things.

01:25 They're all positive.

01:25 There's no subtractions anywhere.

01:27 So n starts at 11 and goes up.

01:30 So i don't need to worry about my absolute value.

01:32 So let me rewrite this in a little bit of an easier form to look at.

01:35 It's 8 to the n plus 1 divided by.

01:39 And now this one is going to be 2n plus 2.

01:42 And i wrote that out just to make sure you see why it's 2n plus 2 factorial.

01:47 And now rather than divided by this, i'm going to multiply by the reciprocal, which is 2n factorial divided by 8 to the n.

01:56 This one was kind of sloppy, but that was eight raised to the n power, not times n there.

02:02 Okay? all right...

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