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 11.4.10

Chapter 11, Problem 11.4.10

Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (sin 2x)/x

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Recall that the Taylor series expansion of \( \sin x \) around \( x = 0 \) is given by \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \).

To find the Taylor series for \( \sin 2x \), substitute \( 2x \) into the series for \( \sin x \), giving \( \sin 2x = 2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \cdots \).

Write the expression \( \frac{\sin 2x}{x} \) using the series expansion: \( \frac{2x - \frac{(2x)^3}{3!} + \cdots}{x} \).

Simplify the fraction by dividing each term in the numerator by \( x \), resulting in \( 2 - \frac{(2x)^3}{3! x} + \cdots \).

Evaluate the limit as \( x \to 0 \) by noting that all terms containing \( x \) vanish, leaving the constant term as the limit.

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Key Concepts

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

Limits

A limit describes the value that a function approaches as the input approaches a certain point. Understanding limits is fundamental in calculus for analyzing behavior near specific points, especially when direct substitution leads to indeterminate forms like 0/0.

Taylor Series

A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. It approximates functions near that point, allowing complex expressions to be simplified and limits to be evaluated more easily.

Evaluating Limits Using Taylor Series

By expanding functions into their Taylor series near the limit point, indeterminate forms can be resolved by canceling terms or simplifying expressions. This method transforms complicated limits into algebraic ones, making it easier to find the limit value.

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Textbook Question

Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).

lim ₓ→₀ (eᵃˣ − 1)/x

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Textbook Question

Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.

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

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Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ (xᵏ/kᵏ)

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Textbook Question

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

∑ₖ₌₀∞ (xᵏ)/(2ᵏ)

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Textbook Question