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Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sin(n \pi /6)}{1 + n\sqrt{n}} $

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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

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Boston College

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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Use any test to determine …

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Determine whether the seri…

Let's show that the following series is absolutely conversion. So to do this instead of looking at the original Siri's, we should look at the absolute value of the terms and then take the son and let me go ahead and rewrite that denominators and to the three over, too. Now we know that sign of any number in absolute value. It's no larger than what so I'LL go here and I'll just replace the new murder with the one and I'LL leave the denominator as it is so you can see him. I'm kind of setting myself up for comparison test due to the inequality that's due to this inequality over here and now this I can go ahead and obtain another upper bound. So let me just go ahead and remove the one in the denominator, and that will make the fraction larger because the new operators were the same. However, the denominator on the right is smaller, so smaller denominator means larger fraction. And now I know this Siri's over here, the Siri's conversions by the Pee test. It's a P Siri's with P equals three over to which is bigger than one, and that's when it converges. And now therefore, because of inequality. This means that our original Siri's no, by the comparison test our cirie, not theory, regional one, but the one that we had when we use absolute value. So by the comparison test this Siri's converges, and by definition it means that the original Siri's is absolutely commercial, So that's our final answer.

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