(a) Use Formula ( 12) to find a series that converges to ln...

(a) Use Formula ( $12)$ to find a series that converges to ln $3 .$
(b) Approximate $\ln 3$ using the first two terms of the series. Round your answer to three decimal places, and compare the result to that produced directly by your calculating utility.

Key Concepts

Partial Sum Approximation

In practice, infinite series are approximated by taking the sum of a finite number of terms, called a partial sum. The quality of the approximation depends on both the number of terms used and the convergence rate of the series. This concept underlies many numerical methods where truncation error is considered in the estimation of values such as logarithms.

Error Estimation and Comparison

When approximating an infinite series with a partial sum, it is important to estimate the error, which is the difference between the true value of the function and its approximation. Comparing the result of a partial sum approximation to a more accurate value (for example, one computed directly by a calculator) provides insight into the effectiveness of the series representation and the rate at which the series converges.

Series Expansion of the Natural Logarithm

This concept involves expressing the natural logarithm function as an infinite series. Particularly, functions such as ln(1+x) or ln((1+z)/(1-z)) have power series representations that can be used to approximate logarithmic values. These series are derived using techniques such as Taylor or Maclaurin series and are valid within a certain radius of convergence.

Convergence of Power Series

A power series representation of a function is only useful if it converges to the function’s actual value for the inputs of interest. Understanding convergence involves determining the interval or radius within which the series sums converge and analyzing the behavior of the remaining (error) terms when truncating the series. This is crucial when applying series approximations to functions like the natural logarithm.

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