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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 11.3.43
Chapter 11, Problem 11.3.43

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.


sinh x²

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Recall the Taylor series expansion for the hyperbolic sine function centered at 0: \(\sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots\).
To find the Taylor series for \(\sinh(x^2)\), substitute \(x^2\) in place of \(x\) in the series for \(\sinh x\). This gives: \(\sinh(x^2) = \sum_{n=0}^{\infty} \frac{(x^2)^{2n+1}}{(2n+1)!} = \sum_{n=0}^{\infty} \frac{x^{4n+2}}{(2n+1)!}\).
Write out the first four nonzero terms explicitly by plugging in \(n=0,1,2,3\) into the series: \(\frac{x^{2}}{1!} + \frac{x^{6}}{3!} + \frac{x^{10}}{5!} + \frac{x^{14}}{7!}\).
Simplify the factorials in the denominators where possible: \(1! = 1\), \(3! = 6\), \(5! = 120\), \(7! = 5040\).
Express the first four nonzero terms of the Taylor series for \(\sinh(x^2)\) as: \(x^{2} + \frac{x^{6}}{6} + \frac{x^{10}}{120} + \frac{x^{14}}{5040} + \cdots\).

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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 derivatives at a single point, usually centered at zero (Maclaurin series). It approximates functions using polynomials, making complex functions easier to analyze and compute.
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Hyperbolic Sine Function (sinh x)

The hyperbolic sine function, sinh x, is defined as (e^x - e^(-x))/2. Its Taylor series expansion at zero includes only odd powers of x with alternating signs, which helps in constructing series for related functions like sinh(x²).
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Substitution in Series Expansions

Substitution involves replacing the variable in a known Taylor series with another expression, such as x², to find the series of composite functions. This technique allows leveraging existing expansions to find new series efficiently.
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Related Practice
Textbook Question

L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.

a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.

b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.

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) = sin x, a = π/2

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Textbook Question

{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

f(x) =∛x with a=64; approximate ∛60.

Textbook Question

Differential equations


a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.


y′(t) − y = 0, y(0) = 2

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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) = 1/(1 - x), a=0

Textbook 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.


1/(1 − 2x)

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