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Find the Maclaurin series of $ f $ (by any method) and its radius of convergence. Graph $ f $ and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and $ f? $

$ f(x) = \tan^{-1} (x^3) $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

Oregon State University

Harvey Mudd College

University of Nottingham

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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Find the Maclaurin series …

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It's a problem. This Find my Clarence theories off off Andi. It's a radio itself. A convergence graph five and its first futon upon Tommy Aslan. Same screen. What do you notice about the relationship between this part? A flux is he got attended The humorist Axe Cube. So first half one over one month Sachs is the call to some from zero between unity, extra ham. Then we have one month or one Prospect Square. This is it was a song negative I squire to the power from a half hand humors X is he called two into girl Blind over one plus x corner Yes, which is he could sum from zero to infinity that you want and Tom's ex too to a path one over to have a swan plus costume number See if we, like actually go to zero we can find C is equal to zero So Tangent universe packs It's a country song from zero to infinity lady you want and have access to it You will ask one over two and plus one. Then after Lux, a sequel to Som from zero to infinity. If you want to end house axe to six and plus three over two and pass one. I will find it's the readers of Convergence Limited Cost. Two zero blank Austrian vanity. I want you off if you want to pass one over two hundred plus three, How's two on us? Want over neck you want? This is a cultural one. So the readers of Convergence Secret who? What now? Let's look at is a graph. Divided curve is a graph ofthe ten and the humerus x Q. The black one this zero but the black oneness to want the blue one is two zero Ondas. We want t two so we can see that at In decreases, he and Max becomes a bite approximation function ofthe attendant X cube.

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