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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.R.65d
Chapter 11, Problem 11.R.65d

Approximating ln 2 Consider the following three ways to approximate
ln 2.
d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.

Verified step by step guidance
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Recall that the natural logarithm function \( \ln(1+x) \) can be expressed as a power series centered at \( x=0 \), given by the Taylor series: \[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
To approximate \( \ln 2 \), recognize that \( \ln 2 = \ln(1 + 1) \). This means we want to evaluate the series at \( x = 1 \).
Substitute \( x = 1 \) into the series to write the infinite series representation for \( \ln 2 \): \[ \ln 2 = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1^n}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \]
This alternating series converges to \( \ln 2 \), and each partial sum provides an approximation of \( \ln 2 \).
Thus, the value of \( x \) to evaluate the series at is \( x = 1 \), and the resulting infinite series for \( \ln 2 \) is the alternating harmonic series as shown above.

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Key Concepts

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

Taylor and Maclaurin Series

Taylor and Maclaurin series express functions as infinite sums of polynomial terms based on derivatives at a specific point. The Maclaurin series is a special case centered at zero. These series allow approximation of functions like ln(x) near a point by summing terms involving powers of (x - a).
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Convergence of Taylor & Maclaurin Series

Choosing the Expansion Point

Selecting the value of x (or the center a) for evaluating a series is crucial for accurate approximation. For ln 2, choosing x close to 1 or another convenient point simplifies the series and improves convergence. The choice affects the form and convergence speed of the infinite series.
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Choosing a Convergence Test

Infinite Series Representation of ln(2)

The natural logarithm ln(2) can be represented as an infinite series derived from the Taylor or Maclaurin expansion of ln(1 + x). For example, ln(2) = ln(1 + 1) can be expressed as the alternating series 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln(2). Understanding this series helps in approximating ln(2) numerically.
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Convergence of an Infinite Series
Related Practice
Textbook Question

Approximating ln 2 Consider the following three ways to approximate

ln 2.

e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?

Textbook Question

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.

ƒ(x) = e^(sin x), n = 2, a = 0

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Textbook Question

Limits by power series Use Taylor series to evaluate the following limits.


lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

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Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series.


sinh (-1)

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Textbook Question

Approximating ln 2 Consider the following three ways to approximate

ln 2.

c. Use the property ln a/b = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for ƒ(x) = ln (1 + x)/(1 - x) b centered at 0.

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Textbook Question

A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.

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