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Use the sum of the first 10 terms to approximate the sum of the series

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{2^n} $

Use Exercise 46 to estimate the error.

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$$s_{10} \approx 1.98828, \quad$ error $\leq 0.012$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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.

02:13

Use the sum of the first 1…

03:30

02:49

03:05

here, we'd like to use the first ten terms. So that will be the tenth partial, some as ten to approximate this infinite sum here from one to infinity. So now, as ten. The first, some of the first ten terms here. So we won't write all these up less used before we would answer this and a calculator. This is what we would end up answering here all the way up to N equals ten. And this would take too long here to write this out by hand. So going to a calculator, approximate this and that That will be one of our answers, the approximation of the infinite son. The question would be, and that's where the air will come in is how close is this approximation to the actual infinite sum? Because ten terms isn't that many. So it might lead you to think that the approximation is not a good one. A question mark there, and that's where the air will come in. So let's not the following that if we define our in to be the consecutive ratio of a N plus one over a n, where this is my end here and then divide by n. And if you take a limit of that as and goes to infinity, this will just become one and then one half. Cancel those two to the end. You just have one over two. Also, Oren is decreasing, decreasing sea ones. One way to show this. So if you look at our inn, we could simplify our previous are and appear before I canceled out that could have been written as and plus one over to end. Now, to show it decreases, you have to either show this or if you define f of X to be X plus one over two and this is usually preferred and works better. In many cases, the second one here show f Prime of X is negative If X is bigger than one, the's are both true here. And so I would recommend just writing this one out and then using the potion rule. So now the reason I'm checking these conditions here is because we want to use exercise number forty six and I'll go to the next page and write this out. So from number forty six, we checked the hypotheses that are needed on our end. So now we can say the following that the remainder after adding ten terms capital Orin is less than or equal to this term here. So this is all due to this previous problem here. So now just using n equals ten and plug it in and we see what we get that gives us our ten and then here will have eleven are eleven. So remember, definition of our end is a and plus one over and which so here we have our eleven, so that will be a twelve over a eleven, and this could be simplified is a fraction where you could just go to the calculator with this is well, and that's the air that we get. So in the previous page, we gave the approximate value. I'Ll recite that here, and the air was on ly this large and that resolves the problem

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