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Find the Taylor polynomials of orders $0,1,2,$ and 3 generated by $f$ at $a$.$$f(x)=\sqrt{1-x}, \quad a=0$$

$$P_{3}(x)=1-\frac{x}{2}-\frac{x^{2}}{4}-\frac{x^{3}}{16}$$

Calculus 2 / BC

Chapter 10

Infinite Sequences and Series

Section 8

Taylor and Maclaurin Series

Series

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

14:11

In mathematics, the partial sums of a series are the sums of all terms of the series except possibly the first and last.

02:24

Find the Taylor polynomial…

03:09

02:40

02:51

For the function f of x, equal to 1 minus x, half 0 is equal to 1. I first derivative is equal to 1 half times 1 minus x to the power of negative 1 half times negative 1 point, so the first derivative at 0 is equal to negative 1. Half second derivative is equal to negative 1 half times negative 1 half to the power times 1 minus x to the power negative 3 over 2 times negative 1. The second derivative at 0 is equal to negative r. Number 42 derivative x is equal to negative 1. Over 4 times negative 3 over 2 times 1 over 1 minus x to the power negative 5 over 2 times negative 1 third tap at 0 is equal to negative 3 over 8. Then the tailor pronominal of that 0 is 1. Of the 1. Is 1 minus half times x? Other 2 is 1 minus 4 half x minus 1 over 4 times x. Squared over 2 factorial of the 3 is 1 minus 1 half x minus an over 4 x squared over 2 factorial minus 3 over 8 times x, cubed over 3 factorial.

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