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.

Verified step by step guidance

1. 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 \(f(x) = \ln(1+x) - \ln(1-x)\).
2. 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 \(|x| < 1\).
3. Substitute the series expansions into the expression for \(f(x)\): \(f(x) = \left( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \right) - \left( -\sum_{n=1}^{\infty} \frac{x^n}{n} \right)\).
4. Simplify the expression by combining the sums: \(f(x) = \sum_{n=1}^{\infty} \left( (-1)^{n+1} + 1 \right) \frac{x^n}{n}\).
5. 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.

Key Concepts

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.

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.

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.