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Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.$ \displaystyle \sum_{n = 1}^{\infty} 5^{-n} \cos^2 n $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Oregon 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.

03:05

Use the sum of the first 1…

02:49

03:30

let's use the some of the first ten terms to approximate the Siri's. So, in other words, this entire series is approximately as ten, and that's equal to the sum from one to ten five, minus in and then co sign swearing in. Now I've approximated this in the calculator already and Wolfram, so you could pause the video, go ahead and write on a few of those decimals. So that's our approximation to the infinite sum. Now this is where we'LL go ahead and actually see what the errors. So first, the best way to go about the year is the first. Think about how you would show this thing convergence. So looking at this is less than or equal to fight to the minus end, says co sign, squared, and is less than or equal to one. And for this series, this converges. You could use the fact that it's geometric. So, for example, for the here. So this is the noted by our ten, and this's bounded above by the air that you would get from using the integral test. So this is explained in more detail on page seven thirty, and then here. If we were to use the integral test. So here this is the integral from ten to infinity, one over five decks. So here you have to show, of course, that the function one over five X So we have to show that it's positive, continuous. These air both clear, and the last one's also clear that is decreasing. It's decreasing because every time X gets bigger, the denominator gets bigger, so the fraction it smaller go ahead and integrate that that's one over five to the ex over Alan off one over five tend to infinity. And so this is approximately point zero zero zero zero three more zeros and then a six. So let's a decimal. And then after the decimal, we have seven zeros, followed by six. So this is an upper bound for an ear, so that gives us an estimate for the year. And that's our final answer.

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