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Determine whether the series is absolutely convergent or conditionally convergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sin n}{n^2} $
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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
Missouri State University
Harvey Mudd College
University of Nottingham
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.
01:39
Determine whether the seri…
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let's go ahead and show that this Siri's is absolutely conversion. So to do that well we should be looking at is not the original Siri's, but the Siri's that you get after you take the absolute value of the terms that your adding. So this here is here and that I just wrote, If this is a convergence series, then the original serious conversions. So if this Siri's converges, then the original is absolutely conversion. Therefore, let's look at this. Siri's here. Now we have sign and over and square. Well, you know, Sign is always less than or equal to one an absolute value. So let's just go ahead and replace sign with one. So here we no sign of end, less than or equal to one that justifies this inequality over here. So we're using the comparison test Now. If we look at this, Siri's here. This Siri's convergence by the PT Test. So it's a Pee series with P equals two that's bigger than one. So it emerges. So by comparison, test Okay, this Siri's due to this inequality over here. That's why we're using comparison. This tells us that if we look at the Siri's of Signe and over and square absolute value from one to infinity that this also converges now going on to the next page. Therefore, we've just shown that the original Siri's from one to infinity sign in over and square that this is absolutely conversion, and that's our final answer.
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