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 11.4.44

Chapter 11, Problem 11.4.44

{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

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Recognize that the integral involves the function \( \frac{\ln(1+t)}{t} \). To approximate this integral using a Taylor series, start by finding the Taylor series expansion of \( \ln(1+t) \) around \( t=0 \).

Recall the Taylor series for \( \ln(1+t) \) is \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{t^n}{n} \) for \( |t| < 1 \). Write this series explicitly: \( \ln(1+t) = t - \frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} + \cdots \).

Divide the series for \( \ln(1+t) \) by \( t \) to get the series for \( \frac{\ln(1+t)}{t} \): \( \frac{\ln(1+t)}{t} = 1 - \frac{t}{2} + \frac{t^2}{3} - \frac{t^3}{4} + \cdots \).

Integrate the series term-by-term from 0 to 0.2: \[ \int_0^{0.2} \frac{\ln(1+t)}{t} dt = \int_0^{0.2} \left(1 - \frac{t}{2} + \frac{t^2}{3} - \frac{t^3}{4} + \cdots \right) dt = \sum_{n=0}^{\infty} a_n \int_0^{0.2} t^n dt, \] where \( a_n \) are the coefficients from the series.

Calculate the integral of each term: \( \int_0^{0.2} t^n dt = \frac{(0.2)^{n+1}}{n+1} \). Sum the terms until the absolute value of the next term is less than \( 10^{-4} \) to ensure the error is within the required tolerance.

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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 function's derivatives at a single point. It allows approximation of complex functions by polynomials, which are easier to integrate or evaluate. For ln(1 + t), the series expansion around t = 0 is especially useful for approximating integrals.

Definite Integral Approximation Using Series

When a function is expressed as a Taylor series, its definite integral over an interval can be approximated by integrating the polynomial term-by-term. This method simplifies the integral calculation and provides a way to estimate the value with controllable accuracy by including enough terms.

Error Estimation and Convergence Criteria

To ensure the approximation is accurate within a specified error bound (e.g., less than 10⁻⁴), it is essential to estimate the remainder or error term of the Taylor series. Understanding how to bound this error helps determine how many terms to retain for a reliable approximation.

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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} 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 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

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

{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

{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

Verified video answer for a similar problem:

Key Concepts

Taylor Series Expansion

Definite Integral Approximation Using Series

Error Estimation and Convergence Criteria

{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