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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{k = 1}^{\infty} (\sin 100)^k $

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$$-0.336$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Baylor University

University of Michigan - Ann Arbor

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.

02:11

Determine whether the seri…

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let's determine whether the series converges or diverges. And then if it converges, we'll go ahead and find the sum. So we see that this is a series of the form like you have. Some number are and it's being raised to the K power. So this is geometric. And in our problem, we see that the are equal sign of 100. It's just a number, and we know geometric will converge only if we have the conditioned. Absolute value. Are is less than one. So really, we want to make sure that we know we know. First of all, that sign satisfies this. It's always between negative one and one, but it can sometimes equal negative one and one. But we would like to make sure that sign of 100 is not equal to plus or minus one. And the reason for this is so that the absolute value of our will not equal one. And the reason that we know that sign of 100 can't be plus or minus one is because of this fact. So let's come over here. If sine X equals one. This means X is equal to pi over two plus two Pi j. And this in a number of this form will never equal 100 because this over here is an irrational number. Whereas 100 or 100 over one, if you want, we can see that that's a fraction of two integers. So this is rational. Yeah. And similarly, if sign equals negative one, this means X is equal to negative pi over two plus two pi j. But once again, we have This can never equal 100 because this is also irrational. So we have that sign of 100. Yeah, is strictly less than one, and therefore this series will converge. Geometric with are less than one and then we'll go ahead and find the sum. That was the second question up here. So the sum let's write this sign 100 to the K. So we know the formula for geometric series. It's that first term, the one that you get by plugging in K equals one in this case, over one minus R. So we have sign of 100. This is after you get you plug in K equals one here and then one minus sign of 100 mm. And this will be the some of the series. So in summary, this given series up here gs geometric, it converges. And down here in the bottom, right, that's the some of the series, and that's our final answer.

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