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

$ f(x) = \sin^2 x $ [Hint: Use $ \sin^2 x = \frac {1}{2} ( 1 - \cos 2x). $ ]

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$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{2 n-1} x^{2 n}}{(2 n) !}, \quad R=\infty$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Oregon State University

Harvey Mudd College

Baylor 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.

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.

09:42

Use a Maclaurin series in …

02:29

03:34

01:56

02:15

01:08

All right, so we can expend this function at the exit, sign a square and the using this hint. All right, so after equals, two, one half over one half times, cousin of two ants. Okay, we're gonna expand, Chosen to X, which is very familiar of us. So one half minus one half times the experiment is going to be one minus two X cube or Victoria plus to excess to hell before or for Victorio. And, of course, it's going to be two extra power for six over six Victoria. And so long. And what is it? You two, which is equal to one half minus one half times so signal. And it's become two to the power to extradite orthe two in over two in factorial. Of course, it's from zero to many. And hear him honest One to power and minus one. Look, actually, basalt. So when in this zero and it becomes too when an issue Oh, no. This is actually could horror in the not minus one minus one. Yeah, this should be correct. We'LL plug it in and equals one. So this is two kings for two is four and three six and Yeah, this is to end and three over. Here it is. Absolute crept end. Obtain the MEChA Lorna series for the function. Okay, so it becomes to ninety one times that you want to our answer me and that you wanted off him. Plus one times So two three power for two months, One extra power to end over to Victorio and the yet for us one half. And we can continue to simplify this. It looks too. And from zero, it's one so infinitely so let you want off this one Two powerful twin minus one times x to in over tutorial plus one half Plus when the zero And when in this zero it becomes too knotty One times that you want times we'LL have. So it's not one half. Why this one half? Yeah, And this kind of hot. So our final answer What is this? Siri's

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