Problem 20 - Thomas Calculus

Question

Find the Maclaurin series for the functions. \(\sinh x=\frac{e^{x}-e^{-x}}{2}\)

Step-by-Step Solution

Answer

The Maclaurin series for \( \sinh x \) is \( \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \).

Step 1: Understand the Function Definition

The function given is expressed as \( \sinh x = \frac{e^{x} - e^{-x}}{2} \). This means we need to express this function in terms of its Taylor series expansion around 0, known as the Maclaurin series.

Step 2: Use the Maclaurin Expansion Formula

The Maclaurin series expansion for a function \( f(x) \) is given by \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \). This requires us to find derivatives of our function at \( x = 0 \).

Step 3: Calculate Derivatives of \(\sinh x\)

First, calculate the derivatives of \( \sinh x = \frac{e^x - e^{-x}}{2} \). The first derivative is \( \cosh x \), the second derivative is \( \sinh x \), and this pattern repeats every two derivatives.

Step 4: Evaluate Derivatives at \( x=0 \)

Evaluate these derivatives at \( x = 0 \): \( \sinh(0) = 0 \), \( \cosh(0) = 1 \), then \( \sinh(0) = 0 \), and so on. Recognize that non-zero terms occur at odd derivatives.

Step 5: Write the Maclaurin Series

Substitute the non-zero coefficients into the Maclaurin series formula. For odd \( n \), the contribution is \( 1 \): The series becomes \( \frac{x}{1!} + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots \).

Step 6: Simplify the Maclaurin Series

Recognize this as \( \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \). This pattern matches our derived terms for \( \sinh x \).

Key Concepts

Taylor Series

Taylor series are incredibly useful in calculus to approximate functions using an infinite sum of terms. The general idea is to expand a function into a power series centered around a specific point, usually zero. This special case, when expanded around zero, is what we call a Maclaurin series.

• A Taylor series for a function \( f(x) \) uses derivatives of the function at a point \( a \).
• The series is expressed as \( \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \).
• For Maclaurin series, \( a = 0 \), so it simplifies to \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \).

The purpose of a Taylor series is to provide a polynomial approximation of the function that simplifies calculus operations and computations. The more terms you include, the more accurate your approximation resembles the original function.

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but for hyperbolas, like how trig functions relate to circles. The most common hyperbolic functions are sinh, cosh, tanh, much like sine, cosine, and tangent in trigonometry.
For the function \( \sinh x \), it's defined as: \[ \sinh x = \frac{e^x - e^{-x}}{2} \] Where \( e^x \) and \( e^{-x} \) are exponential functions. Hyperbolic functions often appear in various areas such as calculus, physics, and engineering, particularly when dealing with waveforms and hyperbolic geometries.

Derivatives

Derivatives tell us how a function changes as its input changes, a fundamental concept in calculus. For the function \( \sinh x \), you take derivatives to build a Taylor or Maclaurin series. here's how it relates:

• The first derivative of \( \sinh x \) is \( \cosh x \), defined as \( \frac{d}{dx}\sinh x = \cosh x = \frac{e^x + e^{-x}}{2}\).
• The second derivative brings us back to \( \sinh x \).
• This cyclical pattern between \( \sinh x \) and \( \cosh x \) continues through higher-order derivatives.

Understanding derivatives helps us evaluate them at specific points, which is critical to develop the Maclaurin series.

Infinite Series

Infinite series are sums of infinitely many terms and are essential for representing mathematical functions and solving various problems. An infinite series takes the form \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) represents the series' terms.
For hyperbolic functions like \( \sinh x \), infinite series are perfect for expressing them in a way that's easier to calculate and integrate. For example, the Maclaurin series for \( \sinh x \) is an infinite alternating series: \[ \sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \] This series uses only odd powers of \( x \), reflecting the odd-derivative values being non-zero. The beauty of infinite series is they allow us to represent functions exactly using polynomials, which are inherently simple to work with.