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What can you say about the series $ \sum a_n $ in each of the following cases?

(a) $ \displaystyle \lim_{n \to \infty} \mid \frac {a_{n + 1}}{a_n} \mid = 8 $

(b) $ \displaystyle \lim_{n \to \infty} \mid \frac {a_{n + 1}}{a_n} \mid = 0.8 $

(c) $ \displaystyle \lim_{n \to \infty} \mid \frac {a_{n + 1}}{a_n} \mid = 1 $

a. Diverges

b. Converges absolutely

c. might converge or it might diverge.

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Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

could we say about the Siri's, the sum of a M and each of the following three cases? So if you look at the left hand side and each of these equations, you can see that they're applying the ratio test. And that says if this limit exist, we have three cases. So has called us Alfa. Oops. So case one we have out for less than one in that case converges absolutely so in particular convergence case to Alfa. Bigger than one in this case, Dan urges. Unfortunately, Case three. This's when Alfa equals one, and the test is what we call inconclusive. In this case, we have to use another method. We just cannot say whether or not a convergence of diverges using this test. So if we look over here in this case, we have eight. That's bigger than one. So in that case, the Siri's were diverge. That's for a party. For Part B. We have less than one point eight. So in that case, the Siri's convergence well, and since they're asking what we can say, let's say as much as we can. We could buy the ratio test. We could say that a convergence absolutely. That's an even stronger statement. So let's go ahead and write that. And in this case, we cannot say anything. So we have Alfa equals one. The ratio test is inconclusive, so we cannot determined weather. It converges there diverges unless we use another method. So it's, um it's important to note that you can still find the convergence or divergent still determine it, but it's just the ratio test itself. I won't tell you the answer when? When the limit is one. So that's why for answer See, there's nothing to conclude but a diver's and being converse, absolutely. So these three answers. It is our final answer.