Explanation Given: The function f ( x ) = cos 2 x , a = 0 . Theorem used: Taylor’s remainder theorem The remainder of the Taylor series is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 for some point c between x and a . Calculation: Obtain the remainder in the Taylor series centered and 0. Substitute 0 for a in R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 , By the theorem stated above, the remainder is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − 0 ) n + 1 = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x ) n + 1 Therefore, the remainder in the Taylor series is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x ) n + 1 for some c . To show that lim n → ∞ R n ( x ) = 0 for all x in the interval of convergence. The interval of convergence for Taylor series cos 2 x is ( − ∞ , ∞ ) . Since the ( n + 1 ) th derivative of cos ( 2 x ) is 2 n + 1 times either ± cos 2 x or ± sin 2 x , | f ( n + 1 ) ( c ) | ≤ 2 n + 1

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.3, Problem 58E

To determine

To find: The remainder in the in the Taylor series centered at a; show that limn→∞Rn(x)=0 for all x in the interval of convergence.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 57E

Chapter 9.3, Problem 59E

Students have asked these similar questions

An+1 3. Find the limit lim for the series 4" n=1 an Does the series converge?

q5 plz provide ans

Represent the function 1x as a power series f(x) Σ 2 + x n=0 Co = C1 = C2 C3 = C4 Find the radius of convergence R = 2

Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

Additional Math Textbook Solutions

Find more solutions based on key concepts

1. On a real number line the origin is assigned the number _____ .

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)

x 2 3x100

Precalculus

Find the slopes of the following lines. The line going through the points (2,5)and(2,8).

Calculus & Its Applications (14th Edition)

Precalculus Enhanced with Graphing Utilities

The temperature at the points x1,x2,x3, and x4 , where each edge of a thin conducting plate is kept at a consta...

Precalculus

Which of the following functions grow faster than ex as x → ∞? Which grow at the same rate as ex? Which grow sl...

Thomas' Calculus: Early Transcendentals (14th Edition)

Knowledge Booster

$Calculus$

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.

The remainder in the in the Taylor series centered at a ; show that lim n → ∞ R n ( x ) = 0 for all x in the interval of convergence.

Solution Summary: The author explains that the remainder in the Taylor series centered at a is R_n(x)=

Videos

To find: The remainder in the in the Taylor series centered at a; show that limn→∞Rn(x)=0 for all x in the interval of convergence.

Want to see the full answer?

Chapter 9 Solutions

Additional Math Textbook Solutions