Homework Help is Here – Start Your Trial Now!
Math
Calculus
Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1

Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

x tan-1 x2

Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1 <xs1 k=1 x x3 x* -In (1 - x) = x + for -1 sx <1 ... %3D k=1 (-1)*x*+1 Σ (-1)* x*+1 tanx = x for |x| s 1 2k + 1 2k + 1 x*+1 x2k+1 Σ for |x| < 0 sinh x = x + 5! (2k + 1)! (2k + 1)!' x* cosh x for |x| < 0 +.. 2! 4! (2k)! (2k)!" moted in Theorem 11.6 the hinoial erieay converge to (1 at +1 denendine on the value of a (3) p(p – 1)(p – 2) - (p – k + 1) (1 + x)P = E()x 2(3) - r*, for x| < 1 and k!
Transcribed Image Text:Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1 <xs1 k=1 x x3 x* -In (1 - x) = x + for -1 sx <1 ... %3D k=1 (-1)*x*+1 Σ (-1)* x*+1 tanx = x for |x| s 1 2k + 1 2k + 1 x*+1 x2k+1 Σ for |x| < 0 sinh x = x + 5! (2k + 1)! (2k + 1)!' x* cosh x for |x| < 0 +.. 2! 4! (2k)! (2k)!" moted in Theorem 11.6 the hinoial erieay converge to (1 at +1 denendine on the value of a (3) p(p – 1)(p – 2) - (p – k + 1) (1 + x)P = E()x 2(3) - r*, for x| < 1 and k!
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Power Series
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS