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The Taylor Series is a powerful mathematical tool used to represent functions as infinite sums of terms calculated from the values of their derivatives at a single point. Specifically, a Maclaurin series is a special case of the Taylor series, centered at zero. For any differentiable function \( f(x) \), the Taylor series expands as:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \)

In the case of the Maclaurin series, where the expansion is about \( a = 0 \), the series simplifies to:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \)

This representation is particularly useful in approximating complex functions with polynomial terms, especially for values of \( x \) close to zero. By following the Taylor series formula, we can systematically derive series for functions like \( \cos x \) and \( \frac{2}{1-x} \), helping us better understand their behavior around the origin.

The geometric series is another essential concept in mathematics, representing a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. A geometric series can be expressed as:

• \( a + ar + ar^2 + ar^3 + \ldots \)

where \( a \) is the first term and \( r \) is the common ratio. When \(|r| < 1\), the series converges to a sum given by:

• \( \frac{a}{1-r} \)

For the function \( \frac{2}{1-x} \), we can expand it using the geometric series formula by recognizing it as:

• \( 2 \cdot \left( 1 + x + x^2 + x^3 + \cdots \right) \)

This expansion is tremendously useful in deriving the Maclaurin series by yielding straightforward expressions for terms in the form of powers of \( x \). Thus, it complements the process of deriving a combined series representation when functions like \( \cos x \) are involved.

The cosine function is a fundamental trigonometric function that can be expanded into an infinite series using the Maclaurin series approach. Known for its even periodic nature, the Maclaurin series for \( \cos x \) is given by:

• \( \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \)

This series results from repeatedly taking derivatives of \( \cos x \) and evaluating them at \( x = 0 \). Because the cosine function is even (symmetric about the \( y \)-axis), the series only contains even-powered \( x \) terms. It provides an accurate approximation of the cosine function near \( x = 0 \), with each additional term improving accuracy.

• For the first three terms, we have: \( 1 - \frac{x^2}{2} + \frac{x^4}{24} \).

When finding the Maclaurin series for composite functions like \( \cos x - \frac{2}{1-x} \), understanding this expansion is crucial. It enables us to systematically subtract or combine terms, as demonstrated in our original exercise, to achieve the desired polynomial approximation.