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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 = e^x \sin^2 x $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Missouri State 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 multiplication of divi…

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Use multiplication or divi…

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The problem is use multiplication of the equation of the power series to find the first 3 nan 0 terms and mc clain series for each function. Y is equal to x into x times sine x, squared so first sine x, squared is equal to 1 minus cosine. 2 x, over 2 police cosine 2 x cosine 2 x is equal to 1 minus 2 x, squared over 2 plus 2 x to the power 4 over 4 factorial minus 2 x to the power of 6 over 6 factorial, plus dot dot dot, which is equal To 1 minus x square plus 2 times x, 24 over 3 in dot dot dot. So we have sine x square is equal to 1 minus cosine 2 x over 2, which is equal to x, squared over 2 minus x, total power 4 over 3 and plus dot dot. Dot e to x is equal to 1 plus x, plus x, squared over 2 plus x, cube over 3 factorial plus dot dot dot on so the half y is equal to into x times sine x. Squared to the first tom is x, squared over 2 point and second term is x. Cubed over 2 third term is x, squared over 2 times x, squared over 2 minus x to the power 4, o 3 plus dot dot dot. This is equal to x, squared over 2 plus x, cube over 2 and next 1 is x to the power 4 over 4 minus x to the power of 4 over 3 pi to this negative x to the power 4 over 12 plus dot dot dot.

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