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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} \frac {12}{(-5)^n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

03:08

Find at least 10 partial s…

04:29

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01:30

$3-8$ Find at least 10 par…

let's find at least ten partial sums of the Siri's. So let's do that first. Well, go ahead and just find as one all the way of Kirsten. Now, as a shortcut, I've used Wolfram to find the cave partial some here and then we're taking Kate to be one all the way up to Capel's ten. So these ten values here in the second column, these Rs one as two three Asfour as five x six a seven as a as nine in s ten. So those air exactly not approximately the first ten sums. Now let's go to the second part saw going back. If you could just pause the screen here to record those values, then I move on to the next part of the question. Now let's grab both the sequence of terms. That's the end civil graft. These guys, so plugging in and equals once attend for an and then we'LL also graph as one to ask him and then we'LL answer the questions. Does it appear that the series is conversion or not? And then we'LL come back and actually find the sum if it is conversion. So now we have a graphing calculator So let me take us that back here. So here I have the sequence A M equals twelve over negative five to the end. So here in purple, this's the graph of the first ten terms. They're very small. There is and gets larger than negative. The denominator. It's very large. And then you could see that it's alternating because of the negative sign, and it looks like the terms were going closer to zero. Now we also have the graph of the partial sums below, and this is will be in a different color. This's in red, so if we scroll down here, we see the first ten terms. So the first value, the first coordinate, is the end value. And then the second corner is the partial, some So here. This's saying that when you add the first six terms, you get about approximately negative one point nine nine nine nine. So it does look like this Red Siri's is converging to negative, too, so let's go ahead and verify this. It looks like the answer is conversion, and let's prove that it is here. The Siri's is geometric, and if that's unclear, you could just go ahead and write the end as negative twelve times negative one over five to the end. So there we see that are our equals negative one over five. This is why income urges. Now let's use the formula for geometric series to find the sum. So the Somme, which I actually just write it out and signal notation from one to infinity, fall over negative five to the end. The formula says you take the first term in the series and then divided by one minus R. So in this case, the first term is negative, twelve over five, and then we divide that by one minus negative, one over five. So this is negative. Twelve over five over six over five, and after cancelling out, we see that that's equals negative, too. So the Siri's converges to negative, too. And that's our final answer.

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