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Use a power series to approximate the definite integral to six decimal places.$ \int^{0.3}_0 \frac {x^2}{1 + x^4} dx $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 9
Representations of Functions as Power Series
Sequences
Series
University of Nottingham
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.
02:54
Use a power series to appr…
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The problem is use a power series to approximate the definite integral to 60 tamal places integral from 0 to 0.3 x, squared over 1 plus x to the power 4. So first we have 1 over 1 plus x. This is equal to some and from 0 to infinity negative 1 to the power of n times x to the power of n. Then we half x, squared over 1 plus x, to the power of 4. This is equal to x squared times, so we use this identity just replace x by x to the power of 4. So we have. This is integral. This is x squared times sum from 0 to infinity negative 1 to the power of n times x, to the power of 4 n, which is equal to some and from 0 to infinity negative 1 times x to the power of 4 n plus 2 point is The integral, from 0 to 0.3 x, squared over 1 plus x to the power of 4 x. This is equal to some from 0 to infinity, to negative 1 to the power n times x to the power of 1 plus 3 over 1 plus 3 from 0. To 0.3, since the n is equal to 3 this term, so we have 0 point 3 to the power of 15 is about 1.43 times 10 to negative 8 point. So we have. This is about 0.3 to the power of 3 over 3 minus 0.3 to the power of the 7 over 7 plus 0.3 to the power of 11 over 11 poi, which is about 0.008969.
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