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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 11.2.44
Chapter 11, Problem 11.2.44

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)

Verified step by step guidance
1
Recall the geometric series formula: \(f(x) = \frac{1}{1 - x} = \sum_{k=0}^{\infty} x^{k}\) for \(|x| < 1\).
To find the power series for \(f(x^{3}) = \frac{1}{1 - x^{3}}\), substitute \(x^{3}\) in place of \(x\) in the original series.
This gives \(f(x^{3}) = \sum_{k=0}^{\infty} (x^{3})^{k} = \sum_{k=0}^{\infty} x^{3k}\).
The power series representation is therefore \(\sum_{k=0}^{\infty} x^{3k}\), which is centered at 0.
Determine the interval of convergence by applying the original condition \(|x| < 1\) to the new variable: since the series is in terms of \(x^{3}\), the condition becomes \(|x^{3}| < 1\), which simplifies to \(|x| < 1\).

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Key Concepts

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

Geometric Series and Its Power Series Representation

A geometric series is a sum of the form ∑ x^k for k from 0 to infinity, which converges to 1/(1-x) when |x| < 1. This fundamental series allows us to express functions as infinite sums, facilitating manipulation and analysis of functions within their radius of convergence.
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Geometric Series

Substitution in Power Series

Substitution involves replacing the variable x in a known power series with another expression, such as x³. This transforms the original series into a new series representing a related function, while maintaining the structure of the series and adjusting the interval of convergence accordingly.
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Intro to Power Series

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges. When substituting variables, the interval changes based on the new expression's magnitude. Determining this interval ensures the validity of the power series representation for the function.
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Related Practice
Textbook Question

{Use of Tech} Remainders Let 


f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ


The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.


a. Show that Rₙ(x) = xⁿ /(1−x).

b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?

c. For fixed n, minimize |Rₙ(x)| with respect to x. Does the result agree with the observations in part (b)?

d. Let N(x) be the number of terms required to reduce |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.

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Textbook Question

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² (ln (1 + t))/t dt

Textbook Question

Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


Find the Taylor polynomial p₃ centered at a=e for f(x)=ln x.

Textbook Question

Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f. What matching conditions are satisfied by the polynomial?

Textbook Question

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

Textbook Question

Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

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