Problem 29 - A History of Mathematics: An Introduction

Question

29\. Calculate the first four nonzero terms of the power series for \(y=\cos z\) using Maclaurin's technique without explicitly using the derivatives of the cosine or sine. Assume that the radius of the circle is 1 .

Step-by-Step Solution

Answer

Answer: The first four nonzero terms of the power series for \(y = \cos z\) are \(1 - \frac{1}{2}z^2 + \frac{1}{24}z^4 + \cdots\).

Step 1: Write down the general formula of the Maclaurin series

The general formula for a function's Maclaurin series is given by: \(y(z) = \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!}z^n\) where \(y^{(n)}(0)\) is the nth derivative of the function evaluated at z=0.

Step 2: Identify the properties of cosine function to help obtain the derivatives

For the cosine function, we know the following properties: 1. \(\cos(0) = 1\) 2. Cosine function is an even function: \(\cos(-z) = \cos(z)\) 3. Its derivatives follow a cyclic pattern, i.e., \((\cos(z))^{(4)} = \cos(z)\)

Step 3: Determine the first four nonzero derivatives of the cosine function evaluated at z=0

Based on the properties mentioned in Step 2, we get the following derivatives evaluated at z=0: 1. \(y^{(0)}(0) = \cos(0) = 1\) 2. \(y^{(1)}(0) = \lim_{h\to 0}\left[\frac{\cos(0+h)-\cos(0-h)}{2h}\right] = \lim_{h\to 0} \left[\frac{\cos(h) - \cos(-h)}{2h}\right] = \lim_{h\to 0} \frac{\sin(h)}{h} = 1\) 3. \(y^{(2)}(0) = \lim_{h\to 0}\left[\frac{\cos(0-h)-\cos(0+h)}{h}\right] = \lim_{h\to 0}\left[\frac{-\cos(h)-\cos(-h)}{2h}\right] = 0\) 4. \(y^{(3)}(0) = \lim_{h\to 0}\left[\frac{\sin(0+h)}{h}\right] = \lim_{h\to 0}\left[\frac{\sin(h)}{h}\right] = 1\)

Step 4: Substitute the derivatives into the general Maclaurin formula and find the first 4 nonzero terms

Using the derivatives obtained in Step 3, substitute these values into the general Maclaurin formula: \(y(z) = \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!}z^n = 1 - \frac{1}{2!}z^2 + \frac{1}{4!}z^4 + \cdots\) The first four nonzero terms of the power series for \(y = \cos z\) are: \( y(z) = 1 - \frac{1}{2}z^2 + \frac{1}{24}z^4 + \cdots\)

Key Concepts

Mathematical Series

In mathematics, a series is an expression composed of the sum of terms of a sequence. Specifically, a mathematical series converges if the sum of its sequence of partial sums tends to a limit. This concept is integral for calculus and higher mathematics, as it allows us to represent complex functions as an infinite sum of simpler terms.

For instance, a Maclaurin series is a type of mathematical series that represents a function as a sum of terms calculated from the values of its derivatives at a single point. It's a special case of a Taylor series, where that point is 0. In the case of trigonometric functions like the cosine function, the Maclaurin series can be particularly useful as it helps simplify these functions into an algebraic form.

Calculus

Calculus is a branch of mathematics focused on rates of change (differential calculus) and the accumulation of quantities (integral calculus). Underpinning a vast range of scientific disciplines, it allows the description of change, motion, and growth.

When we deal with functions like the cosine function in calculus, we often resort to differentiation — finding derivatives — to understand how the function behaves. The concept of the derivative is used to build the power series for a function, as seen in the calculation of the Maclaurin series for cosine. This series helps us to predict the value of cosine for very small angles, without relying on the trigonometric angle knowledge, which can be immensely helpful in fields like physics or engineering.

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 of the nth term, and \(x\) is a variable. It's a powerful tool used in calculus to express functions as an infinite sum of their terms.

Power series are central to the concept of analytic functions, which are functions that can be represented by a power series in some interval or disk around any point in their domain. The Maclaurin series, for the cosine function or otherwise, is one such power series where the point of expansion is zero, meaning \(x=z\) in this context. This expression offers a practical way to approximate functions close to the point of expansion using polynomial terms.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental within mathematics, defining relationships between the angles and sides of triangles. They also extend beyond triangles and find applications in modeling periodic phenomena such as waves and oscillations.

Understanding the cosine function and its properties is crucial in solving many mathematical problems. Its Maclaurin series expansion represents the cosine function as a sum of powers of variable \(z\), revealing the function's underlying structure. This aids in calculating angles without explicitly resorting to trigonometric tables or the unit circle. For students learning calculus, grasping how trigonometric functions can be transformed into an infinite series of algebraic terms opens the door to a deeper comprehension of both the functions themselves and the mathematical world they model.