Skip to main content
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.R.62
Chapter 11, Problem 11.R.62

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. There is more than one way to choose the center of the series.


sinh (-1)

Verified step by step guidance
1
Recall the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
Use the Taylor series expansion of \(\sinh x\) centered at 0 (Maclaurin series), which is given by \(\sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}\).
Write out the first four nonzero terms of this series explicitly: \(x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \frac{x^{7}}{7!}\).
Substitute \(x = -1\) into the series to approximate \(\sinh(-1)\), remembering to keep the powers and factorials intact without calculating the numerical values.
Express the resulting series as \(-1 + \frac{(-1)^{3}}{3!} + \frac{(-1)^{5}}{5!} + \frac{(-1)^{7}}{7!}\), which are the first four nonzero terms of the infinite series for \(\sinh(-1)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 derivatives at a single point called the center. It approximates functions near this center, allowing complex functions to be expressed as polynomials. Choosing the center wisely can simplify calculations and improve convergence.
Recommended video:
08:42
Taylor Series

Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It is an odd function with a well-known Taylor series expansion around zero, involving only odd powers of x. Understanding its properties helps in constructing its series representation accurately.
Recommended video:
5:53
Graph of Sine and Cosine Function

Finding Nonzero Terms in Series

When approximating functions using Taylor series, identifying the first few nonzero terms is crucial for an accurate approximation. This involves computing derivatives at the center and recognizing which terms vanish due to the function's symmetry or properties, ensuring the series reflects the function's behavior.
Recommended video:
06:00
Geometric Series
Related Practice
Textbook Question

Limits by power series Use Taylor series to evaluate the following limits.


lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

1
views
Textbook Question

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.


∞

Σ x⁴ᵏ/k²

k = 1

1
views
Textbook Question

Approximating ln 2 Consider the following three ways to approximate

ln 2.

d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.

Textbook Question

Approximating ln 2 Consider the following three ways to approximate

ln 2.

c. Use the property ln a/b = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for ƒ(x) = ln (1 + x)/(1 - x) b centered at 0.

1
views
Textbook Question

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)


ƒ(x) = eˣ; bound R₃(x), for |x| < 1

Textbook Question

A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.

1
views