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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.3.61
Capitolo 11, Problema 11.3.61

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series


(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.


1/(3 + 4x)²

Guida verificata passo dopo passo
1
Start with the given Maclaurin series for \((1 + x)^{-2}\): \[ (1 + x)^{-2} = 1 - 2x + 3x^{2} - 4x^{3} + \cdots \]
Rewrite the function \(\frac{1}{(3 + 4x)^2}\) in a form that resembles \((1 + u)^{-2}\). Factor out the constant from the denominator: \[ \frac{1}{(3 + 4x)^2} = \frac{1}{9(1 + \frac{4x}{3})^2} \]
Identify the substitution \(u = \frac{4x}{3}\) so that the function becomes: \[ \frac{1}{9} (1 + u)^{-2} \]
Use the known Maclaurin series for \((1 + u)^{-2}\) and substitute \(u = \frac{4x}{3}\) into the series: \[ (1 + u)^{-2} = 1 - 2u + 3u^{2} - 4u^{3} + \cdots \]
Multiply the entire series by \(\frac{1}{9}\) and expand the terms up to the \(x^{3}\) term to find the first four nonzero terms of the Maclaurin series for \(\frac{1}{(3 + 4x)^2}\).

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Maclaurin Series

A Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of terms involving powers of x and derivatives evaluated at zero. Understanding this allows approximation of functions near zero using polynomials.
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Convergence of Taylor & Maclaurin Series

Power Series Substitution and Manipulation

Power series substitution involves replacing the variable in a known series with an expression to find a new series expansion. Factoring and algebraic manipulation help rewrite functions into forms compatible with known series, enabling term-by-term expansion.
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Intro to Power Series

Binomial Series Expansion

The binomial series generalizes the expansion of expressions like (1 + x)^n for any real exponent n. It provides coefficients for each term in the series, which is essential for expanding functions like (1 + x)^{-2} and related forms within their radius of convergence.
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Geometric Series
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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.


f(x³) = 1/(1 − x³)

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


∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k

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Tangent line is p₁ Let f be differentiable at x=a


a. Find the equation of the line tangent to the curve y=f(x) at (a, f(a)).


b. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).

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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.

tan ⁻¹ (1/2)

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Power series for derivatives


a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.


f(x) = ln (1 + x)

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{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.


√(1+x) ≈ 1 + x/2 on [−0.1,0.1]

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