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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 11.2.60
Capitolo 11, Problema 11.2.60

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x) = ln √(1 − x²)

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1
Rewrite the function to a simpler form using logarithm properties: \(f(x) = \ln \sqrt{1 - x^2} = \frac{1}{2} \ln(1 - x^2)\).
Recall the known power series expansion for \(\ln(1 - u)\) centered at 0, which is \(\ln(1 - u) = -\sum_{n=1}^{\infty} \frac{u^n}{n}\) for \(|u| < 1\).
Substitute \(u = x^2\) into the series to get \(\ln(1 - x^2) = -\sum_{n=1}^{\infty} \frac{x^{2n}}{n}\), valid for \(|x^2| < 1\) or \(|x| < 1\).
Multiply the entire series by \(\frac{1}{2}\) to match the original function: \(f(x) = \frac{1}{2} \ln(1 - x^2) = -\frac{1}{2} \sum_{n=1}^{\infty} \frac{x^{2n}}{n}\).
State the interval of convergence based on the substitution and original series: since \(|x| < 1\), the interval of convergence is \((-1, 1)\).

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Power Series Representation

A power series is an infinite sum of terms in the form a_n(x - c)^n, where a_n are coefficients and c is the center. Representing functions as power series allows approximation and analysis using polynomials. Finding a power series for a function often involves manipulating known series or using derivatives and integrals.
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Intro to Power Series

Known Power Series and Manipulation

Common functions like ln(1+x), 1/(1-x), and sqrt(1-x) have established power series expansions. To find the series for a related function, we use substitution, algebraic manipulation, or differentiation/integration of these known series. For example, ln(√(1 - x²)) can be expressed using the series for ln(1 - x²).
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Intro to Power Series

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges to the function. It depends on the radius of convergence and endpoint behavior. Determining this interval ensures the series accurately represents the function within that domain.
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Interval of Convergence
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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. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series


√(1 + x) = 1 + x/2 − x²/8 + x³/16 − ⋯, −1 ≤ x ≤ 1.


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Differential equations


a. Find a power series for the solution of the following differential equations, subject to the given initial condition

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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.


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Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

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