Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

One can sometimes find a Maclaurin series by the method of equating coefficients. For example, let $$ \tan x=\frac{\sin x}{\cos x}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots $$ Then multiply by \(\cos x\) and replace \(\sin x\) and \(\cos x\) by their series to obtain $$ \begin{aligned} x-\frac{x^{3}}{6}+\cdots &=\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\left(1-\frac{x^{2}}{2}+\cdots\right) \\ &=a_{0}+a_{1} x+\left(a_{2}-\frac{a_{0}}{2}\right) x^{2}+\left(a_{3}-\frac{a_{1}}{2}\right) x^{3}+\cdots \end{aligned} $$ Thus, $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}-\frac{a_{0}}{2}=0, \quad a_{3}-\frac{a_{1}}{2}=-\frac{1}{6}, \quad \cdots $$ so $$ a_{0}=0, \quad a_{1}=1, \quad a_{2}=0, \quad a_{3}=\frac{1}{3}, \quad \cdots $$ and therefore $$ \tan x=0+x+0+\frac{1}{3} x^{3}+\cdots $$ which agrees with Problem 1 . Use this method to find the terms through \(x^{4}\) in the series for \(\sec x\).

Step-by-Step Solution

Answer

\( \sec x = 1 + \frac{x^2}{2} + \frac{x^4}{3} + \cdots \)

Step 1: Express the Series

We want to find the Maclaurin series for \( \sec x \), which is expressed as \( \sec x = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots \).

Step 2: Multiply by Cosine Series

Multiply \( \sec x = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots \) by the cosine series \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \) to get: \[1 = (a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \cdots)(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots)\]

Step 3: Expand and Equate Coefficients

Expand the right-hand side of the equation and compare coefficients with the left-hand side, which is 1:\[1 = a_0 + a_1 x + \left(a_2 - \frac{a_0}{2}\right)x^2 + \left(a_3 - \frac{a_1}{2}\right)x^3 + \left(a_4 - \frac{a_2}{2} + \frac{a_0}{24}\right)x^4 + \cdots\]No terms of \(x\) or higher appear on the left side.

Step 4: Solve for Coefficients

From the equations:- \( a_0 = 1 \)- \( a_1 = 0 \)- \( a_2 - \frac{1}{2} = 0 \rightarrow a_2 = \frac{1}{2} \)- \( a_3 = 0 \)- \( a_4 - \frac{1}{2} + \frac{1}{24} = 0 \rightarrow a_4 = \frac{1}{3} \)Thus, the coefficients are: \( a_0 = 1, a_1 = 0, a_2 = \frac{1}{2}, a_3 = 0, a_4 = \frac{1}{3}\).

Step 5: Write the Maclaurin Series

Substitute back into the series form: \( \sec x = 1 + 0\cdot x + \frac{1}{2} x^2 + 0\cdot x^3 + \frac{1}{3} x^4 + \cdots \).

Key Concepts

Taylor Series

The Taylor Series is a mathematical tool used to represent a wide variety of functions as infinite sums of terms calculated from the values of their derivatives at a single point. It helps in approximating complex functions by polynomials, which are easier to compute. The Taylor series of a function \( f(x) \) about a point \( a \) is given by:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \]Depending on the desired degree of accuracy, you truncate the series after a certain number of terms. When the Taylor series is expanded around \( x = 0 \), it is known as the Maclaurin Series, which is particularly useful for functions like \( \sin x \), \( \cos x \), and \( \tan x \).

• Explains complex functions with simpler polynomials.
• Has applications in engineering, physics, and computer science.
• The most common case, the Maclaurin series, represents function centered around zero.

Series Expansion

Series expansion is a method of writing a function as an infinite sum of terms calculated from its derivatives at a particular point. This technique assists in approximating a function through polynomials of increasing degree. The process starts by finding the respective derivatives of the function and evaluating them at a specific point, often zero in the case of Maclaurin series.

For instance, given a function \( f(x) \), its series expansion around zero uses terms like \( f(0) \), \( f'(0)x \), \( \frac{f''(0)}{2!}x^2 \), and so on. This expansion makes it feasible to appraise functions that are otherwise difficult to compute precisely. It serves as a foundational concept in calculus and is crucial for numerical methods implementations.

• Translates complicated functions into a series of easily calculable terms.
• Used extensively in solving differential equations and mathematical modeling.
• Helps in numerical computations and approximations in computer algorithms.

Trigonometric Series

Trigonometric series, especially the series expansions for functions like \( \sin x \), \( \cos x \), and \( \tan x \), are tools used to represent trigonometric functions as infinite sums. These series are integral in calculus for solving many real-world problems. Such series help in deriving approximation formulas for trigonometric expressions needed in physics and engineering.

• For \( \sin x \): \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \)
• For \( \cos x \): \( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \)
• For \( \tan x \), the series can be derived similarly once \( \sin x \) and \( \cos x \) have been expanded.

These series are essential when digital devices need to compute trigonometric functions efficiently and accurately without using traditional trigonometric tables.

Calculus

Calculus is the mathematical study that focuses on understanding change. It involves two major branches: differential calculus and integral calculus. Differential calculus studies the rate of change and slopes of curves. Integral calculus deals with the accumulation of quantities and areas under or between curves. The Taylor and Maclaurin series fall under the umbrella of calculus as they involve the derivatives of functions.

Calculus is essential in fields such as physics, engineering, economics, statistics, and many others. The Maclaurin series and other series expansions are specific applications of calculus concepts, allowing infinite sums to approximate functions.

• Provides foundational tools for describing dynamic systems and changes.
• Applications range from computing complex integrals to solving differential equations.
• Used in optimizing functions and systems in economics and operational research.

The power of calculus ensures that we can understand and predict a wide array of natural phenomena and engineered systems.