Find the first four nonzero terms of the Maclaurin series...

Find the first four nonzero terms of the Maclaurin series for the function by using an appropriate trigonometric identity or property of logarithms and then substituting in a known Maclaurin series.
$$\text { (a) } \cos ^{2} x \quad \text { (b) } \ln \left(\frac{1-x}{1+x}\right)$$

Key Concepts

Maclaurin Series

A Maclaurin series is a type of Taylor series expansion of a function about zero. It expresses a function as an infinite sum of its derivatives at 0 multiplied by powers of x. Understanding how to determine the coefficients and the radius of convergence is essential for approximating functions or extracting specific terms in a power series.

Trigonometric Identities

Trigonometric identities, such as the double-angle formula, are used to simplify complex trigonometric expressions before expansion. For instance, an identity like cos²x = (1 + cos2x)/2 can reduce the problem of expanding a squared cosine function to that of expanding a linear combination of constants and a cosine, which has a well?known series expansion.

Logarithmic Properties

Properties of logarithms, such as the quotient rule ln(a/b) = ln a - ln b, allow one to break down complex logarithmic expressions into simpler parts. This simplification is particularly useful when each resulting term has an established Maclaurin series, making it easier to determine the combined series expansion.

Known Power Series Expansions

Familiarity with the standard power series expansions of elementary functions like cosine and natural logarithm is crucial. These known series, such as the cosine series and the logarithmic series ln(1+u) = u - u²/2 + u³/3 - ... (for |u| < 1), serve as building blocks when constructing the Maclaurin series for more complicated functions through substitution and algebraic manipulation.

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