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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.R.49
Chapter 11, Problem 11.R.49

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


lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

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Recall the Taylor series expansion of \( \cos x \) around \( x = 0 \): \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]
Substitute the Taylor series expansion of \( \cos x \) into the given expression: \[ \frac{\frac{x^2}{2} - 1 + \cos x}{x^4} = \frac{\frac{x^2}{2} - 1 + \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \right)}{x^4} \]
Simplify the numerator by combining like terms: \[ \frac{x^2}{2} - 1 + 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots = \frac{x^4}{24} - \cdots \]
Rewrite the expression with the simplified numerator: \[ \frac{\frac{x^4}{24} - \cdots}{x^4} \]
Divide each term in the numerator by \( x^4 \) and then take the limit as \( x \to 0 \). The higher order terms vanish, leaving the constant term from the division.

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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 like cosine to be expressed as polynomials, which simplifies limit evaluation.
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Limit Evaluation Using Series

By substituting the Taylor series into the limit expression, we can rewrite the limit in terms of powers of x. This method helps identify dominant terms and simplifies the limit calculation, especially when direct substitution leads to indeterminate forms.
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Handling Indeterminate Forms

Limits that result in forms like 0/0 require algebraic manipulation or series expansion to resolve. Using power series expansions transforms the expression into a form where the limit can be directly computed by canceling terms or evaluating the leading coefficients.
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Related Practice
Textbook Question

Approximating ln 2 Consider the following three ways to approximate

ln 2.

e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?

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

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


sinh (-1)

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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.

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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.

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