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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 11.2.76
Chapter 11, Problem 11.2.76

{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.

Verified step by step guidance
1
Step 1: Understand the definitions given. The function is defined as the infinite geometric series \(f(x) = \sum_{k=0}^\infty x^k = \frac{1}{1-x}\) for \(|x| < 1\). The partial sum up to \(n\) terms is \(S_n(x) = \sum_{k=0}^{n-1} x^k\). The remainder after \(n\) terms is \(R_n(x) = f(x) - S_n(x)\).
Step 2: To show that \(R_n(x) = \frac{x^n}{1-x}\), start from the formula for the partial sum of a geometric series: \(S_n(x) = \frac{1 - x^n}{1 - x}\). Then express the remainder as \(R_n(x) = f(x) - S_n(x) = \frac{1}{1-x} - \frac{1 - x^n}{1 - x}\). Simplify this expression to isolate \(R_n(x)\).
Step 3: For the graphing part, consider the function \(R_n(x) = \frac{x^n}{1-x}\) on the interval \(|x| < 1\) for \(n=1, 2, 3\). Plot these functions to observe how the remainder behaves as \(x\) varies. Pay attention to the magnitude \(|R_n(x)|\) and note where it reaches its maximum and minimum values within the interval.
Step 4: To minimize \(|R_n(x)|\) for fixed \(n\), analyze the function \(|R_n(x)| = \left| \frac{x^n}{1-x} \right|\). Consider taking the derivative with respect to \(x\) and setting it to zero to find critical points. Check these points within the domain \(|x| < 1\) to determine where the minimum occurs. Compare these results with the observations from the graph.
Step 5: For the function \(N(x)\), which is the number of terms needed to make \(|R_n(x)| < 10^{-6}\), set up the inequality \(\left| \frac{x^n}{1-x} \right| < 10^{-6}\). Solve for \(n\) in terms of \(x\), which will give \(N(x)\). Then graph \(N(x)\) over the interval \(|x| < 1\) to visualize how many terms are required for different values of \(x\).

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Key Concepts

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

Geometric Series and Partial Sums

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. The infinite series ∑ₖ₌₀∞ xᵏ converges to 1/(1−x) for |x|<1. The partial sum Sₙ(x) sums the first n terms and approximates the infinite sum, serving as a foundation for understanding series truncation and remainders.
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Intro to Series: Partial Sums

Remainder (Error) of a Power Series Approximation

The remainder Rₙ(x) measures the difference between the infinite series and its partial sum after n terms. For the geometric series, Rₙ(x) = xⁿ/(1−x), quantifying the truncation error. Understanding this remainder helps analyze convergence behavior and error bounds in approximations.
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Intro to Power Series

Minimization and Graphical Analysis of Functions

Minimizing |Rₙ(x)| involves finding values of x that reduce the remainder's magnitude, often using calculus or inspection. Graphing Rₙ(x) over |x|<1 reveals where the error is largest or smallest, providing visual insight into convergence rates and guiding the choice of n for desired accuracy.
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Related Practice
Textbook Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₁∞ (x²ᵏ)/k

Textbook Question

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

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

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)

Textbook Question

{Use of Tech} Graphing Taylor polynomials


a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n=1 and n=2.


b. Graph the Taylor polynomials and the function.


f(x)=sin x, a=π/4