Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems 43-48 to see that you get the same answers using the methods of Section 9.7. $$ (\sin x) /(\exp x) $$

Step-by-Step Solution

Answer

The first four nonzero terms are: \( x, -\frac{x^3}{3}, \frac{x^5}{10}, -\frac{x^7}{42} \).

Step 1: Write the expressions for the Maclaurin series

The Maclaurin series expansion provides the Taylor series of a function centered at 0. For sine and exponential functions, the expansions are well-known. The Maclaurin series for \( \sin x \) is: \[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]The Maclaurin series for \( \exp x \) is:\[\exp x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\]

Step 2: Divide the series

To find the Maclaurin series for \( \frac{\sin x}{\exp x} \), you need to divide the series term by term.Using the series from Step 1, the division for the first few terms is:\[\frac{\sin x}{\exp x} = \frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots}{1 + x + \frac{x^2}{2!} + \cdots}\]

Step 3: Perform the division using CAS

Enter the expression into a Computer Algebra System (CAS) to perform the division. The CAS will give the result as a series:\[\frac{\sin x}{\exp x} \approx x - \frac{x^3}{3} + \frac{x^5}{10} - \frac{x^7}{42} + \cdots\]

Step 4: Collect the first four nonzero terms

From the result obtained from the CAS, extract the first four nonzero terms. In this case, they are:- First term: \( x \)- Second term: \(-\frac{x^3}{3} \)- Third term: \(\frac{x^5}{10} \)- Fourth term: \(-\frac{x^7}{42} \)

Step 5: Verify with decomposition

Verify the correctness by checking with manual decomposition (as seen in Section 9.7). Perform long division if necessary to ensure these terms are correct.

Key Concepts

Taylor Series

The Taylor series is a fundamental concept in calculus, providing a way to represent functions as infinite sums of terms calculated from the derivatives at a single point. Specifically, the Maclaurin series is a special case of the Taylor series centered at zero. For any function \( f(x) \), its Taylor series can be expressed as:

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

Here, \( f^{(n)}(0) \) denotes the \( n \)-th derivative of \( f(x) \) evaluated at \( x = 0 \). This series provides an approximation of the function around the center point, which in the case of a Maclaurin series, is 0. When calculating a function's Maclaurin series, we often use only the first few terms, as they provide a reasonable approximation with simpler calculations, especially useful for small values of \( x \).

Sine Function Series

The sine function, \( \sin(x) \), can be expanded into its Maclaurin series, which is one of the classic examples of such expansions. This series helps us understand how the sine function behaves through a polynomial approximation. The series is expressed as:

• \( \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \)

Each term in the series incorporates the odd powers of \( x \) and factorial denominators, alternating in sign. Truncating the series to the first few terms provides an efficient method to estimate the sine function's value for small angles, where it behaves almost linearly with the input. The beauty of the sine function series lies in its simplicity and the insights it offers into periodic functions expanded into polynomials.

Exponential Function Series

The exponential function \( \exp(x) \), or \( e^x \), has a very neat and tidy Maclaurin series expansion. This series is extremely useful in mathematics and engineering, due to its universally convergent property for all real \( x \). The Maclaurin series for the exponential function is written as:

• \( \exp(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \)

This expansion shows that all terms are positive and involve every power of \( x \) divided by the factorial of its index, capturing the idea of continuous growth without bounds. The exponential series is crucial in solving differential equations and in modeling growth processes. Its role in approximating \( e^x \) and its involvement in complex computations signal its importance in mathematical analysis.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a powerful tool used to perform symbolic mathematics. This includes tasks like expanding functions into their series, manipulating algebraic expressions, and even solving equations analytically. In the context of Maclaurin series, a CAS efficiently handles the tedious process of term-by-term division and ensures precision that is sometimes difficult to achieve manually.

• Allows accurate series expansion and manipulation.
• Quickly provides results for complex computations.
• Helpful for verifying manual calculations in textbook problems.

When using a CAS for dividing two series, such as \( \frac{\sin(x)}{\exp(x)} \), it automates division operations and produces a series expansion directly, highlighting its usefulness in both educational and professional settings. The right use of CAS can significantly enhance understanding and precision in working with complex mathematical concepts.