Approximating ln 2 Consider the following three ways to approximate
ln 2.
b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.

Verified step by step guidance

1. Recall the Taylor series expansion for \( \ln(1 - x) \) centered at 0, which is given by the infinite series: \[ \ln(1 - x) = -\sum_{n=1}^{\infty} \frac{x^n}{n} \quad \text{for} \quad |x| < 1. \]
2. Use the identity \( \ln 2 = -\ln \frac{1}{2} \). This means we can write \( \ln 2 \) as the negative of \( \ln(1 - x) \) by choosing \( x \) such that \( 1 - x = \frac{1}{2} \).
3. Solve for \( x \) in the equation \( 1 - x = \frac{1}{2} \), which gives \( x = \frac{1}{2} \).
4. Substitute \( x = \frac{1}{2} \) into the Taylor series for \( \ln(1 - x) \) to express \( \ln \frac{1}{2} \) as: \[ \ln \frac{1}{2} = \ln(1 - \frac{1}{2}) = -\sum_{n=1}^{\infty} \frac{(\frac{1}{2})^n}{n}. \]
5. Finally, apply the negative sign from the identity \( \ln 2 = -\ln \frac{1}{2} \) to write \( \ln 2 \) as the infinite series: \[ \ln 2 = -\ln \frac{1}{2} = \sum_{n=1}^{\infty} \frac{(\frac{1}{2})^n}{n}. \]

Key Concepts

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. For ln(1 - x) centered at 0, the series expresses ln(1 - x) as a power series in x, which converges for |x| < 1. This allows approximation of ln(1 - x) by summing a finite number of terms.

Logarithm Properties and Identities

Logarithm identities, such as ln(a/b) = ln a - ln b and ln(1/x) = -ln x, help transform expressions into more convenient forms. Using ln 2 = -ln(1/2) leverages these properties to rewrite ln 2 in terms of ln(1 - x), enabling the use of the Taylor series for ln(1 - x) to approximate ln 2.

Convergence of Infinite Series

Understanding the convergence criteria of the Taylor series is essential to ensure the approximation is valid. The series for ln(1 - x) converges when |x| < 1, so choosing x = 1/2 fits within this radius. Recognizing convergence guarantees that the infinite series accurately represents ln(1 - x) and thus ln 2.