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Use series to approximate the definite integral to within the indicated accuracy.

$ \int^1_0 \sin (x^4) dx $ $ \text { (four decimal places)} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

Harvey Mudd College

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.

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Use series to approximate …

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05:57

The problem is your series to approximate definite integral to within indicated accuracy integral from 0 to 1 sine x to 4 x, that is a to integral from 0 to 1 sum from 0 to infinity negative 1 to n times x, 24 to the power 2 n, Plus 1, over 2 n plus 1 factorial ex, which is equal to some that from 0 to infinity negative 1 to n times x, 8 n plus 5 over 8 n plus 5 times 2 n plus 1 factorial from 0 to 1 point. This is equal to some from 0 to infinity negative 1 to n times 1 over 8 n plus 5 times 2 n plus 1 factorial. My end is equal to 3 half life over portenone has 7 factorial is about 0.000007. So behalf, this series is about 1. Over 5 minus 1 over 13 times 3 factorial plus 1 over 21 times 5 factorial, which is about 0.1 ei.

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