Problem 17 - Calculus: One and Several Variables

Question

Expand \(g(x)\) as indicated and specify the values of \(x\) for which the expansion is valid. \(g(x)=\cos x \quad\) in powers of \(x-\pi\).

Step-by-Step Solution

Answer

The Taylor series expansion of \(g(x) = \cos x\) around \((x-\pi)\) is \(-1 + \frac{(x-\pi)^2}{2!} - \frac{(x-\pi)^4}{4!} + \cdots\). The expansion is valid for all real numbers.

Step 1: Rewrite the function

We can rewrite \(g(x) = \cos(x)\) as \(g(x) = \cos(x - \pi + \pi)\). This is done because we want to express the function in terms of \((x - \pi)\).

Step 2: Use the cosine difference formula

The cosine difference formula states: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). So we can rewrite \(\cos(x - \pi + \pi)\) as \(\cos(x - \pi)\cos\pi + \sin(x - \pi)\sin\pi\).

Step 3: Substitute the values

Substituting \(\cos\pi = -1\) and \(\sin\pi = 0\) results in \(-\cos(x - \pi)\). This represents the equivalent expression for \(g(x) = \cos x\) in terms of \((x - \pi)\).

Step 4: Expand in powers of \(x-\pi\)

The Taylor series expansion of \(\cos x\) in powers of \(x - a\) is \(\cos a - (x-a)\sin a - \frac{(x-a)^2}{2!}\cos a + \frac{(x-a)^3}{3!}\sin a + \cdots\). Substituting \(a = \pi\), and changing \((x-\pi)\) in the expansion, gives us \(-1 - (x-\pi)0 - \frac{(x-\pi)^2}{2!}(-1) + \frac{(x-\pi)^3}{3!}0 - \cdots\). Therefore the series reduces to \(-1 + \frac{(x-\pi)^2}{2!} - \frac{(x-\pi)^4}{4!} + \cdots\).

Step 5: Determine the validity of the expansion

The expansion is valid for all \(x\) since the domain of the cosine function is all real numbers.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is an important trigonometric function that is often encountered in mathematics. It describes the relationship between the angle and the length of the adjacent side in a right-angled triangle, relative to the hypotenuse. Cosine is a periodic function with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) radians.
The cosine function takes an input angle \( x \) (in radians), and outputs a value ranging from \(-1\) to \(1\). This makes it a bounded function. The cosine of zero, \( \cos(0) \), equals \(1\). As the angle increases towards \( \pi \) radians, the cosine value decreases to \(-1\) and then ascends back to \(1\) as the angle reaches \( 2\pi \).
In mathematical series and transformations, the cosine function serves as a base for expanding functions into series, enabling deeper analysis and easier computation.

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) represents the coefficients of the series, \( c \) is the center of the expansion, and \( x \) is the variable.
Power series are incredibly valuable in mathematics because they can represent complex functions in a form that is easy to manipulate, solve, or approximate. In the context of the cosine function, we use a Taylor series expansion to represent \( \cos(x) \) around a specific point \( c \).
For \( \cos(x) \), the Taylor series expansion at \( x = \pi \) transforms the cosine function into a polynomial form involving powers of \((x - \pi)\). For example, when expanded, the series from the problem is \(-1 + \frac{(x-\pi)^2}{2} - \frac{(x-\pi)^4}{24} + \cdots\), capturing the behavior of the cosine function around the point \( \pi \). This helps simplify calculations and analysis of the function's behavior around that neighborhood.

Domain of Validity

In mathematics, the domain of validity of a series expansion refers to the set of values for which the series accurately represents the function. For trigonometric functions like the cosine function, it's important to determine where the series provides a true reflection of the original function's output.
The Taylor series expansion of \( \cos(x) \) is valid for all real numbers. This is derived from the nature of the cosine function, which itself is defined and continuous for all real inputs. Essentially, no matter what real value \( x \) takes, the series converges to the actual value of \( \cos(x) \).
For the given exercise, we expand \( \cos(x) \) in terms of \( x-\pi \), and note that this particular series can be used for all real \( x \) to approximate \( \cos(x) \) as closely as desired by including more terms in the series, affirming its very broad domain of validity.