Approximating ln 2 Consider the following three ways to approximate
ln 2.
a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.

Verified step by step guidance

1. Recall the Taylor series expansion for \( \ln(1 + x) \) centered at 0 (Maclaurin series), which is given by the infinite sum: \[ \ln(1 + x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \quad \text{for} \quad -1 < x \leq 1. \]
2. To approximate \( \ln 2 \), substitute \( x = 1 \) into the series because \( \ln 2 = \ln(1 + 1) \). This gives: \[ \ln 2 = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1^n}{n} = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}. \]
3. Write out the first few terms explicitly to see the pattern: \[ \ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots \]
4. Note that this is an alternating series where the terms decrease in magnitude and approach zero, which ensures convergence to \( \ln 2 \).
5. This infinite series representation is the exact Taylor series expression for \( \ln 2 \) centered at 0, and it can be used to approximate \( \ln 2 \) by summing a finite number of terms.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. For ln(1 + x) centered at 0, the series expresses ln(1 + x) as a power series in x, allowing approximation by partial sums. This method is fundamental for approximating functions near the center point.

Convergence of Infinite Series

Convergence refers to whether the infinite sum of a series approaches a finite value. For the Taylor series of ln(1 + x) at x = 1, convergence ensures the series sum equals ln(2). Understanding the interval and conditions of convergence is essential to justify using the series for approximation.

Natural Logarithm Function Properties

The natural logarithm ln(x) is the inverse of the exponential function e^x and is defined for x > 0. Its behavior near 1 is smooth and differentiable, making it suitable for Taylor expansion. Recognizing properties of ln(1 + x) helps in setting up and interpreting the series approximation.