Approximate the specified function value as indicated and...

Approximate the specified function value as indicated and check your work by comparing your answer to the function value produced directly by your calculating utility.
Approximate tan $^{-1} 0.1$ to three decimal-place accuracy using the Maclaurin series for $\tan ^{-1} x$

Key Concepts

Maclaurin Series

The Maclaurin series is a specific type of Taylor series centered at zero. It expresses a function as an infinite sum of terms calculated from the derivatives of the function at zero. By approximating the function with a finite number of terms, one can efficiently estimate values near the expansion point, which is particularly useful when the function itself is complicated to compute directly.

Arctan Function Series

The Maclaurin series for the arctan function is derived from its Taylor series expansion centered at zero. It takes the form x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ..., where the series alternates in sign and includes only odd-powered terms. This series is valid for |x| ? 1 and provides a method for approximating arctan values by summing enough terms until the desired precision is achieved.

Error Estimation and Accuracy

When using series approximations, it's important to estimate the error introduced by truncating the series. Error estimation involves determining how many terms are needed so that the difference between the true value and the approximated value is within acceptable bounds, such as three decimal-place accuracy. This ensures that the approximation is not only computationally efficient but also reliably close to the actual function value.

Convergence of Series

The convergence of a series defines the range of values for which the series will converge to the actual function value. For the arctan series, convergence is guaranteed when |x| ? 1. Understanding the convergence criteria is crucial because it confirms that the series approximation will produce valid and accurate results within the specified range of x.

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