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Find the Maclaurin series for $ f(x) $ using the definition of a Maclaurin series. [ Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0. $] Also find the associated radius of convergence.

$ f(x) = (1 - x)^2 $

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

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Harvey Mudd College

University of Nottingham

Idaho State University

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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Find the Maclaurin series …

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so fun. The connoisseurs for FX using a definition of mech Lauren Siri's all right. So the definition of it is going to be. And from zero to infinity, the doing of half at zero. Over in factorial, Tom's ex called in. Okay, so after X equals two Woman is a square. So after zero equals one and I front of X on at zero, it's going to be on two times women's likes and no one at zero. So this is just Teo minus two and the second your video a zero is going to be so this part, it's ah, pleased to X minus two. Do it here and Cyril in just two. And you see the first, the first third directive is gonna be zero and so on is going to zero. So you have only three times for every thanks. The first time is some one times extreme help zero, which is one class is minus two axe blows. Teo Victoria Square, which is just one wants to expose. It's where so you see, we just expand this square, their equivalent Oh, the horses. Just the whole realize of this infinity

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