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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.65c
Chapter 11, Problem 11.R.65c

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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Recall the given function: \(f(x) = \ln \left( \frac{1+x}{1-x} \right)\). Using the logarithm property \(\ln \frac{a}{b} = \ln a - \ln b\), rewrite \(f(x)\) as \(f(x) = \ln(1+x) - \ln(1-x)\).
Identify the Taylor series expansions centered at 0 for \(\ln(1+x)\) and \(\ln(1-x)\) separately. These are standard series: \(\ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}\) for \(|x| < 1\), and \(\ln(1-x) = -\sum_{n=1}^{\infty} \frac{x^n}{n}\) for \(|x| < 1\).
Substitute the series expansions into the expression for \(f(x)\): \(f(x) = \left( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \right) - \left( -\sum_{n=1}^{\infty} \frac{x^n}{n} \right)\).
Simplify the expression by combining the sums: \(f(x) = \sum_{n=1}^{\infty} \left( (-1)^{n+1} + 1 \right) \frac{x^n}{n}\).
Analyze the term \((-1)^{n+1} + 1\) to determine which terms survive in the series. This will help you write the final Taylor series for \(f(x)\) centered at 0.

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Key Concepts

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

Properties of Logarithms

Logarithmic properties, such as ln(a/b) = ln(a) - ln(b), allow us to rewrite complex logarithmic expressions into simpler forms. This is essential for breaking down functions like ln((1+x)/(1-x)) into differences of ln(1+x) and ln(1-x), facilitating the use of known series expansions.
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Change of Base Property

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, often centered at zero. Understanding how to find and manipulate Taylor series is crucial for approximating functions like ln(1+x) and ln(1-x) near x=0.
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Taylor Series

Combining Series for Function Operations

When a function is expressed as a combination of other functions, their Taylor series can be combined term-by-term. For ln((1+x)/(1-x)), this means subtracting the series of ln(1-x) from ln(1+x), enabling the derivation of the Taylor series for the quotient function.
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Geometric 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

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.


ƒ(x) = 1/(1 - x²)

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.

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.