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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.4.47
Chapter 11, Problem 11.4.47

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
cos 2

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Recall the Taylor series expansion of the cosine function centered at 0 (Maclaurin series): \[\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\]
To approximate \( \cos 2 \), substitute \( x = 2 \) into the series expansion: \[\cos 2 = 1 - \frac{2^2}{2!} + \frac{2^4}{4!} - \frac{2^6}{6!} + \cdots\]
Identify the first four nonzero terms of this series explicitly: 1st term: \( 1 \) 2nd term: \( - \frac{2^2}{2!} \) 3rd term: \( + \frac{2^4}{4!} \) 4th term: \( - \frac{2^6}{6!} \)
Write the partial sum of these four terms as the approximation for \( \cos 2 \): \[\cos 2 \approx 1 - \frac{2^2}{2!} + \frac{2^4}{4!} - \frac{2^6}{6!}\]
This expression represents the first four nonzero terms of the infinite Taylor series for \( \cos 2 \). You can stop here or calculate each factorial and power if a numerical approximation is desired.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point, usually around zero (Maclaurin series). It allows approximation of functions like cosine by polynomials, making complex functions easier to analyze and compute.
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Maclaurin Series for Cosine

The Maclaurin series for cosine is a specific Taylor series centered at zero, given by cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ... . This alternating series includes only even powers of x and is used to approximate cosine values near zero.
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Convergence of Taylor & Maclaurin Series

Term Selection and Approximation Accuracy

When approximating functions using Taylor series, selecting the first few nonzero terms balances simplicity and accuracy. Including more terms improves precision but increases complexity. For cos(2), using the first four nonzero terms provides a good polynomial approximation of the value.
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Related Practice
Textbook Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = e⁻ˣ, a = 0

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (eˣ − e⁻ˣ)/x

Textbook Question

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.


g(x) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)

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Textbook Question

Radius of convergence Find the radius of convergence for the following power series.

∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)

Textbook Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₀∞ (xᵏ)/(2ᵏ)

Textbook Question

Combining power series Use the power series representation


f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,


to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.


p(x) = 2x⁶ ln(1 − x)