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Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function.

$ y = (\arctan x)^2 $

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$x^{2}-\frac{2 x^{4}}{3}+\frac{23 x^{6}}{45}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Missouri State University

Campbell University

Oregon 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.

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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 multiplication or divi…

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Use multiplication of divi…

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The problem is use multiplication of division of power series to find the first 3 and 0 terms in the maclaine series for each function. Y is equal to 10 in the x squared. So first we half tangent x is equal to x, minus x, cube over 3 plus x 25 over fif minus x, 7 over 7 plus dot dot dot, and then it times x, minus x, cube over 3 plus x to the power of over half minus x. To 7 over 7 start to do, but the first terms x times e magnatum is x times negative x, cube over 3 is x times negative x, cubed over 3 plus negative x, cube over 3 times x, and then the third term is x as x to The power 5 over 5 minus plus x cube over 3 times x, cube over 3 plus x to the power 5 over 5 times as to do. This is equal to x, square minus 2 third times x to the power 4 plus. This is 23 over 45 times x to its power of 6, plus dot dot, dot.

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