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.

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Recall that the natural logarithm function \( \ln(1+x) \) can be expressed as a power series centered at \( x=0 \), given by the Taylor series: \[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]

To approximate \( \ln 2 \), recognize that \( \ln 2 = \ln(1 + 1) \). This means we want to evaluate the series at \( x = 1 \).

Substitute \( x = 1 \) into the series to write the infinite series representation for \( \ln 2 \): \[ \ln 2 = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1^n}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \]

This alternating series converges to \( \ln 2 \), and each partial sum provides an approximation of \( \ln 2 \).

Thus, the value of \( x \) to evaluate the series at is \( x = 1 \), and the resulting infinite series for \( \ln 2 \) is the alternating harmonic series as shown above.

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Taylor and Maclaurin Series

Taylor and Maclaurin series express functions as infinite sums of polynomial terms based on derivatives at a specific point. The Maclaurin series is a special case centered at zero. These series allow approximation of functions like ln(x) near a point by summing terms involving powers of (x - a).

Choosing the Expansion Point

Selecting the value of x (or the center a) for evaluating a series is crucial for accurate approximation. For ln 2, choosing x close to 1 or another convenient point simplifies the series and improves convergence. The choice affects the form and convergence speed of the infinite series.

Infinite Series Representation of ln(2)

The natural logarithm ln(2) can be represented as an infinite series derived from the Taylor or Maclaurin expansion of ln(1 + x). For example, ln(2) = ln(1 + 1) can be expressed as the alternating series 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln(2). Understanding this series helps in approximating ln(2) numerically.

Approximating ln 2 Consider the following three ways to approximateln 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.