Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.4.51

Chapter 11, Problem 11.4.51

Evaluating an infinite series Let f(x) = (eˣ − 1)/x, for x ≠ 0, and f(0)=1. Use the Taylor series for f centered at 0 to evaluate f(1) and to find the value of ∑ₖ₌₀∞ 1/(k+1)!

Verified step by step guidance

Recognize that the function is given by \(f(x) = \frac{e^{x} - 1}{x}\) for \(x \neq 0\), and \(f(0) = 1\). Our goal is to find the Taylor series of \(f\) centered at 0, then use it to evaluate \(f(1)\) and the infinite sum \(\sum_{k=0}^{\infty} \frac{1}{(k+1)!}\).

Recall the Taylor series expansion of \(e^{x}\) centered at 0: \(e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}\). Substitute this into the expression for \(f(x)\) to write \(f(x)\) as a series:

\[f(x) = \frac{e^{x} - 1}{x} = \frac{\sum_{n=0}^{\infty} \frac{x^{n}}{n!} - 1}{x} = \frac{\sum_{n=1}^{\infty} \frac{x^{n}}{n!}}{x}.\]

Simplify the fraction by dividing each term in the numerator by \(x\): \(f(x) = \sum_{n=1}^{\infty} \frac{x^{n}}{n!} \cdot \frac{1}{x} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{n!}\). Reindex the sum by letting \(k = n - 1\), so that:

\[f(x) = \sum_{k=0}^{\infty} \frac{x^{k}}{(k+1)!}.\]

To find \(f(1)\), substitute \(x=1\) into the series: \(f(1) = \sum_{k=0}^{\infty} \frac{1^{k}}{(k+1)!} = \sum_{k=0}^{\infty} \frac{1}{(k+1)!}\). This shows that the value of the infinite sum \(\sum_{k=0}^{\infty} \frac{1}{(k+1)!}\) is equal to \(f(1)\), which can be evaluated using the original function definition.

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Key Concepts

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

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. For a function f(x) centered at 0, it is expressed as f(x) = Σ (f⁽ᵏ⁾(0)/k!) xᵏ. This allows approximation and evaluation of functions like f(x) = (eˣ − 1)/x near zero.

Maclaurin Series for the Exponential Function

The Maclaurin series for eˣ is Σ (xᵏ / k!) from k=0 to ∞. Using this, eˣ − 1 can be written as Σ (xᵏ / k!) for k=1 to ∞. Dividing by x shifts the index, which helps express f(x) as a power series and evaluate it at specific points like x=1.

Infinite Series Summation and Evaluation

Evaluating infinite series involves recognizing patterns and using known series sums. Here, the series ∑ₖ₌₀∞ 1/(k+1)! corresponds to the sum of reciprocals of factorials shifted by one index, which relates to the exponential function minus 1. Understanding this helps find exact values of such sums.

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Related Practice

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.

f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)

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 polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ

Textbook Question

Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.

Textbook Question

{Use of Tech} Approximations with Taylor polynomials

a. Approximate the given quantities using Taylor polynomials with n = 3.

b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.

e⁰ᐧ¹²

Textbook Question

{Use of Tech} Approximations with Taylor polynomials

a. Approximate the given quantities using Taylor polynomials with n = 3.

b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.

√1.06

Textbook Question

{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.

a. Find p₅ and q₅

b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?

c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

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