Problem 26 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\cos x-\sin x$$

Step-by-Step Solution

Answer

Taylor series: \(1 - x - \frac{x^2}{2} + \frac{x^3}{6} + \ldots\)

Step 1: Recall Taylor Series for Cosine and Sine

Begin by writing down the known Taylor series expansions for \(\cos x\) and \(\sin x\) centered at \(x=0\). The Taylor series for \(\cos x\) is \(\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\) and for \(\sin x\) is \(\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\).

Step 2: Subtract the Series

Subtract the series for \(\sin x\) from the series for \(\cos x\). Start with the expressions:\[ \cos x - \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} - \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}. \]

Step 3: Combine the Series into a Single Series

To combine the two series, align the terms based on powers of \(x\). For even powers of \(x\), terms come from \(\cos x\), and for odd powers of \(x\), terms come with a negative sign from \(\sin x\). Thus the series becomes:\[ \sum_{n=0}^{\infty} \left( \frac{(-1)^n x^{2n}}{(2n)!} - \frac{(-1)^n x^{2n+1}}{(2n+1)!} \right). \]

Step 4: Simplify the Expression

Simplify the expression by factoring out common terms, if possible, and evaluating initial terms to see the pattern. The series starts as follows: \(1 - x - \frac{x^2}{2} + \frac{x^3}{6} + \ldots\).

Step 5: Write Final Taylor Series

Consolidate your findings to write out the final Taylor series:\[ \cos x - \sin x = \sum_{n=0}^{\infty} \left( \frac{(-1)^n x^{2n}}{(2n)!} - \frac{(-1)^n x^{2n+1}}{(2n+1)!} \right). \]

Key Concepts

Power Series

A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) are the coefficients and \( c \) is the center of the series. When dealing with Taylor series, the center often is chosen as 0, simplifying the series to \( \sum_{n=0}^{\infty} a_n x^n \). Power series are useful in approximating functions and can provide highly accurate results over a particular domain.

• The convergence of a power series depends on the value of \( x \) and its distance from the center \( c \).
• The series is characterized by its radius of convergence; within this radius, the series converges to a function.
• Power series can be added, subtracted, and integrated term by term as long as they share the same interval of convergence.

Taylor series are power series that are specifically used to represent functions around a specific point. For functions like cosine and sine, these series give an easy way to compute approximate values for different inputs.

Cosine Function

The cosine function, denoted as \( \cos x \), is an even trigonometric function, which means \( \cos(-x) = \cos(x) \). This symmetry makes it ideal for Taylor series representation, especially since its series consists only of even powers of \( x \). This makes calculations slightly simpler in practice.
The Taylor series for \( \cos x \) centered at \( x=0 \) is given by:\[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\]

• Each term in the series alternates in sign, indicated by \((-1)^n\).
• The term involves only even powers, \( x^{2n} \), reflecting the even nature of the cosine function.
• The factorial \((2n)!\) in the denominator grows quickly, which influences the convergence speed of the series.

By using this series, cosine values can be calculated to a reasonable degree of precision without needing calculator functions.

Sine Function

The sine function, represented as \( \sin x \), is an odd trigonometric function, which means \( \sin(-x) = -\sin(x) \). This characteristic is reflected in its Taylor series, which only contains odd powers of \( x \). This structure provides a straightforward way to approximate \( \sin x \) for small values of \( x \).
The Taylor series for \( \sin x \) centered at \( x=0 \) is:\[\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\]

• Similar to the cosine series, the sign of the terms alternate due to \((-1)^n\).
• This series involves only odd powers \( x^{2n+1} \), aligning with the odd nature of the sine function.
• The factorial \((2n+1)!\) also contributes to the convergence characteristics of the series by keeping larger terms small.

The representation of sine through a power series makes it easier to understand and use in various mathematical and engineering applications without needing precise numerical computations of \( \sin x \) directly.