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

$ \int^{0.4}_0 \sqrt{1 + x^4} dx $ $ \left( \mid \text {error} \mid < 5 \times 10^{-6} \right) $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Oregon State University

Harvey Mudd College

Idaho State University

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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$51 - 54$ Use series to ap…

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The problem is your series to approximate it even integral to wetting that indicated accuracy integral from 0 to 0.4 times. Square root of 1 plus x to fort is less than 5 times 10 to negative 6 point. The first to be half square root of 1 plus x 24 is equal to some and from 0 to 21, half to n times 4. The integral, from 0 to 0.4 square of 1 plus x for the x, is equal to some and from 0 to infinity or half n times x, 24 n plus 1 over 4 n plus 1 from 0 to 0.4, which is equal to some from 0 to Infinity the half times and at the half n times, 0.424 n plus 1 over 4 n plus 1 feeds on half in is equal to the half is negative 1 half his negative 3 half is negative. 5. Over 2 times to the dog is 1 half minus n minus 1 over in factorial. So we can see that what is greater than 1. This series is an alternating series by the alternating series in estimation. Seriem behalf are involved in this approximation or 4 n plus 1 is like less than 0.424 n plus 5. Over 4. Here is a value 4 plus 5 times you times, half n plus 1. So by n is equal to 2. Have r is 9, is less than 1 half 3 times 0.4 to the power 13 over 13, which is about 3.23 times 10 to negative 8, which is less than 5 times 10 to negative 6 point. So we have, this: series is about 0.4 over 1 plus 1 half 1 times 0.425 over 5 plus 1, half 2 times 0.429 over 9, which is about 0.40102 zerox.

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