Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\tanh x $$

Step-by-Step Solution

Answer

The Maclaurin series for \( f(x) = \tanh x \) through \( x^5 \) is \( x - \frac{x^3}{3} + \frac{2x^5}{15} \).

Step 1: Understand the Given Function

The given function is the hyperbolic tangent, \( f(x) = \tanh x \). A hyperbolic tangent can be defined as \( \tanh x = \frac{\sinh x}{\cosh x} \). So, our task is to find the terms through \( x^5 \) in the Maclaurin series for \( \tanh x \).

Step 2: Review Known Maclaurin Series

Recall the Maclaurin series expansions for the hyperbolic sine and cosine. The series for \( \sinh x \) is \( x + \frac{x^3}{3!} + \frac{x^5}{5!} + \ldots \), and for \( \cosh x \) is \( 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \ldots \). These will help us derive the series for \( \tanh x \).

Step 3: Divide the Series of \(\sinh x\) and \(\cosh x\)

Express the division \( \tanh x = \frac{\sinh x}{\cosh x} \) using the series already known. Start by writing the series of \( \sinh x \) as \( x + \frac{x^3}{6} + \frac{x^5}{120} \) and \( \cosh x \) as \( 1 + \frac{x^2}{2} + \frac{x^4}{24} \). Therefore, divide the series of \( \sinh x \) by \( \cosh x \).

Step 4: Long Division of Series

Perform long division between the series \( x + \frac{x^3}{6} + \frac{x^5}{120} \) and \( 1 + \frac{x^2}{2} + \frac{x^4}{24} \) to find the series of \( \tanh x \). Divide step by step to first approximate \( x \), then calculate higher powers of \( x \) up to \( x^5 \).

Step 5: Calculate Terms Through \( x^5 \)

Through long division, find the first few terms of the series: \( x - \frac{x^3}{3} + \frac{2x^5}{15} + \ldots \). This gives us the terms through \( x^5 \) for the Maclaurin series of \( \tanh x \).

Step 6: Verify and Write the Final Series

Review the steps to ensure accuracy. The final series for \( \tanh x \) through \( x^5 \) is \( x - \frac{x^3}{3} + \frac{2x^5}{15} \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but for a hyperbola instead of a circle. Two primary hyperbolic functions are the hyperbolic sine, written as \( \sinh x \), and the hyperbolic cosine, written as \( \cosh x \). These functions are defined as:

• \( \sinh x = \frac{e^x - e^{-x}}{2} \)
• \( \cosh x = \frac{e^x + e^{-x}}{2} \)

The hyperbolic tangent, \( \tanh x \), is another crucial hyperbolic function, expressed as the ratio of \( \sinh x \) to \( \cosh x \): \[ \tanh x = \frac{\sinh x}{\cosh x} \]These functions model real-world phenomena, like the shape of cables hanging under their own weight, known as a catenary curve. Understanding hyperbolic functions is vital for learning higher-level calculus and differential equations.
Without these, exploring the dynamics of certain physical systems would be nearly impossible.

Series Expansion

Series expansion is a powerful mathematical technique that represents functions as infinite sums of terms. This method is very useful for approximating complicated functions using polynomials. The Maclaurin series is a specific type of series expansion based at 0 (a simpler form of Taylor series) that provides approximations by adding higher powers of \( x \) sequentially:
\[ f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \]For the function \( \tanh x \), the process involves:

• Using known expansions for simpler functions such as \( \sinh x \) and \( \cosh x \)
• Combining these expansions to achieve the series expansion of \( \tanh x \)
• Finding the terms up to a certain power of \( x \) to simplify calculations

Applying this concept helps reduce complex functions to manageable, polynomial approximations, which are easier to work with and provide insights into function behavior near \( x = 0 \).

Calculus Techniques

In tackling problems such as deriving the Maclaurin series for \( \tanh x \), it is important to use calculus techniques effectively. This includes:

• Finding derivatives of functions at a point, particularly at \( x = 0 \)
• Utilizing well-known series expansions like the exponential function for efficiently solving division or multiplication problems involving series
• Being familiar with algebraic manipulation, such as expanding series through multiplication, division, and seeking patterns in terms of powers of \( x \)
• Performing long division on polynomial series which can often provide a practical route toward the desired series terms

Mastering these techniques is essential for simplifying complex expressions and provides a systematic approach to various types of calculus problems. These skills are transferable across different areas of mathematics, emphasizing their importance in a mathematician's toolkit.

Power Series

A power series is an infinite series of the form \[\sum_{n=0}^{\infty} a_n x^n\]where \( a_n \) represents the coefficient for the \( n \)-th term and \( x^n \) is the power of \( x \). Power series serve as a foundation for series expansions like the Maclaurin series.
These series allow the approximation of functions around a point by using polynomial terms. Considerations for using power series include:

• Convergence: Understand where the series converges, meaning where it accurately represents the function it approximates
• Manipulation: Knowing how to add, subtract, and perform other operations on series to transform and simplify expressions
• Application: Power series can model various phenomena, from simple motion equations to complex fluid dynamics algorithms

The versatility of power series enables their use across mathematics, serving a broad audience from high school students to engineers and physicists. Additionally, grasping power series is pivotal for deeper studies in mathematical analysis and other advanced subjects.