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Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.

$ f(x) = x \cos \left( \frac {1}{2} x^2 \right) $

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02:03

Wen Zheng

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

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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 a Maclaurin series in …

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01:46

So with the McLaren Siris in Table One, we know that the coastline of acts is equal to the sum from an equal zero to infinity of negative one to the end, Times X to the two n over two and factorial. And we could write that out if we want. But it's gonna be one minus x squared over two factorial plus X to the fourth over four factorial minus X to the sixth over six sectorial. So that's gonna keep going on. And we have a radius of convergence of infinity, Then to get X co sign X squared over two. What we're gonna do is just replace every instance of X with X squared over two and then we'll multiply the whole thing by X. So what we end up getting as a result is going to be the summation from an equal zero to infinity of negative one to the end of exports by the thing by X Times X squared to the to end over to to the two n times two and factorial. Then we could simplify this further. If we want. That would ultimately just end up giving us and equal zero to infinity negative one to the end, X to the four and plus one over to to the two n two n factorial. Then, with that in mind, we want to use ratio test. So we're gonna look at the absolute value of a M plus one over a n. When we do that, we end up getting X to the fourth over four times to n plus 2/2 n plus one. Then we want to take the limit of both of those as n goes to infinity. So when we take the limit of that, we end up getting zero. So because of that, the Siri's we know will converge for all X, which means that the radius of convergence is infinity.

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