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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 $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 6
Absolute Convergence and the Ratio and Root Tests
Sequences
Series
Oregon State University
Baylor University
University of Michigan - Ann Arbor
Lectures
01:59
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.
02:28
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.
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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.
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