Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Use a power series to approximate the definite integral to six decimal places.$ \int^{0.2}_0 x \ln (1 + x^2) dx $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Carson Merrill

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

04:25

Wen Zheng

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Missouri State University

Harvey Mudd College

Idaho State University

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.

02:50

Use a power series to appr…

02:57

02:54

04:28

04:27

02:15

So what we have here is the integral from 0 to 0 to x times the natural log of one plus X squared DX, um, and the Power Series. What we have is the summation. This is the natural log of one plus X equals the summation of an equals one to infinity of negative one to the end, minus one of X over and corrects to the end over. And if we replace X with X squared, what we end up getting is the summation from n equals one to infinity of negative one to the N minus one of times X to the two n over end. So then when we multiply by X, what we get here when we multiply by X that's going to give us. Um, since we're doing that, it's going to end up giving us again the summation. So we can just copy that portion right here. The only thing that's gonna be changing is what's in the top. So as a result, we can duplicate this and this is just going to have a plus one right here. Then we can integrate this, um, integrating the summation. What? This is gonna look like is just now this from 0 to 0 to, um, and recall that will be integrate. Now, this is going to be, um, two and plus two. And and this is going to be X to the two n plus two. Um, Then what we end up getting is that this is going to equal summation from zero to infinity. We're sorry and equals one to infinity of negative one to the N minus one of 0.2 to end plus two over two and plus two times n minus zero. If we look at how and how what happens as an increases, we see that the term is, uh, very small. So we'll use the alternating series estimation therapy. So we have is that s minus two absolute value. That is less than ankle to be three, which is less than zero point 000 000107 So knowing this, we just need to add all the terms before B three, and what we end up getting as a result is that this is going to end up giving us approximately 0.0 395

View More Answers From This Book

Find Another Textbook