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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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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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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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