Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.4.11

Chapter 11, Problem 11.4.11

Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − e⁻ˣ)/x

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Recall the Taylor series expansions of the exponential functions around 0: \(e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\) and \(e^{-x} = 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots\).

Substitute these expansions into the expression \(\frac{e^{x} - e^{-x}}{x}\) to get \(\frac{\left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\right) - \left(1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots\right)}{x}\).

Simplify the numerator by combining like terms: the constant terms cancel out, and you combine the \(x\) terms, \(x^{2}\) terms, etc., carefully noting the signs.

After simplification, factor out \(x\) from the numerator to cancel with the denominator, which will help in evaluating the limit as \(x\) approaches 0.

Evaluate the resulting expression by substituting \(x = 0\) into the simplified series to find the limit.

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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. It approximates functions near that point, allowing complex expressions to be simplified into polynomials. For example, eˣ can be expanded around x=0 as 1 + x + x²/2! + x³/3! + ....

Limit Evaluation Using Series

When direct substitution in a limit leads to an indeterminate form, expanding functions into their Taylor series can simplify the expression. By substituting series expansions, terms can often be canceled or simplified, making it easier to find the limit as x approaches a point.

Hyperbolic Sine Function and Its Relation

The expression (eˣ − e⁻ˣ)/2 defines the hyperbolic sine function, sinh(x). Recognizing this helps simplify the limit problem since (eˣ − e⁻ˣ)/x = 2 * (sinh(x)/x). Understanding sinh(x) and its series expansion aids in evaluating limits involving exponential differences.

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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

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

Textbook Question

Combining power series Use the geometric series

f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,

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

g(x) = x³/(1 − x)

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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

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)