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Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.$ \displaystyle \sum_{n = 1}^{\infty} \cos n $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

University of Michigan - Ann Arbor

University of Nottingham

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.

07:39

Find at least 10 partial s…

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$3-8$ Find at least 10 par…

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01:30

let's find atleast ten partial sums of the Siri's. So to do that, we'LL just find as one as two always Teo as ten These are the first ten partial zones and so I'll actually do that and graft them at the same time. So we go to my graphing calculator and Dez Mose And then here let me a raise this and graph iss So here are my first ten sums the first cordon and is telling you the value of end. So here here's s one and then the value is point five four and so on and going all the way up to the S ten, which is about negative one point four one seven. So here you can pause this screen and the right coordinates are giving you the first ten values. So now that's a graph of the sequence of partial sums. So we already graft that, and on the graph, we saw the values of the partial sums. So then there's another part Here were to graft the sequence of terms. So here would. They also would like to see a one a two all the way to a ten where Anne is just coastline of n. So let's also graph in a different color. Let me a race the graph for a moment of the partial sums and let me put in the graph of the ends. So here the co sign values Excuse me? Not yet. So there we go and purple. We see the graph of the co signs. I have a label there. If you want deposit, you could record those right hand values. Otherwise, I'll just remove the labels. And if we just look at this purple graph, we can see that it doesn't look like the values will converge. And if you like, I'LL go ahead and you can plot both the sequence and and purple and the partial sums and red. So coming back to the original problem does it appear that the Siri's convergence or diverges by the graph it looks like it diverges. So I'll go back to the graph and read. We can see that it doesn't really look like the values are approaching anything, so that's a guess. But if you want a precise answer, you can show the limit of co sign, and it doesn't exist, so cannot be zero. So the Siri's diverges bye the divergence test, and that's our final answer

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