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Approximate the sum of the series correct to four decimal places. $ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^n ne^{-2n} $

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$-0.1050$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Missouri State University

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.

03:27

Approximate the sum of the…

02:25

01:09

03:09

01:23

21:16

Find the sum of the series…

05:11

02:47

01:18

Let's approximate the Siri's correct up to four decimal places. So first of all, we see that this is alternating. So here one would use the alternating Siri's estimation Kerem. But before this is used, what we would have to show that the serious convergence. So the Siri's we can show that a conversions by the alternating Siri's serum. So here, to do that, we can define B end to just be the positive part of the A. M. So just this part right here. These terms are both positive, so this is clearly positive, the limit being equal zero. This is true, and you can use low Patel's rule here to show this, and we can also show that the be ends are decreasing. If you want to show this, you can just define ffx to be ex e negative to X, and then show that the door evidence is negative, so that means it's decreasing. So by the alternating Siri's test alternating serious narrow, this series will converge. And now, by the alternating Siri's estimation there on this tells us that we can have a bound on the ear. This is less than our people to be in plus one. So the ear is when we approximate the entire Siri's, but only using in terms. Now this is by the serum. Moreover, if we want correct to four decimal places, we shouldn't make the air. This law, I'LL put four zeros in a five. So if we have any number of less than this, that means that there is not large enough to affect the fourth decimal spot, because when you round off, it will not change the zero into a one because the fifth decimal will be less than five. So this is why I used this inequality at the end. So using this and up here, we'd like to know when and e to the negative to end is less than this. So go on to the next page, using a calculator that turns out to be when N is bigger than six. But really, we wanted be an plus one not being so really we want and plus one bigger than or equal to six, and I gives us and to be five or more. So this means that we'Ll approximate using five terms. In other words, as if we just do as five. There's some music just of the first five terms, so I could even write that out. This is just the sum and equals one toe. Five negative, one to the end and e to the minus to end. Now we're on ly going toe five places. Here are four places. Excuse me. So I get negative zero point one zero five zero and that's correct. Up to four decimals, and that's our final answer.

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