Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.R.62

Capitolo 11, Problema 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)

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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 \(\sinh(-1)\).

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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.

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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.

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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.

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Geometric Series

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Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.

ƒ(x) = 1/(1 - x²)

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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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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.

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