Problem 24 - Calculus Single Variable

Question

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=x \cos (x) $$

Step-by-Step Solution

Answer

The Maclaurin series for \(f(x) = x \cos(x)\) is \[\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n)!}.\]

Step 1: Recall the Maclaurin Series for Cosine

The Maclaurin series for the function \(\cos(x)\) is given by: \[\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}.\] This series is expanded about \(x=0\) and involves only even powers of \(x\).

Step 2: Multiply Each Term by x

To find the Maclaurin series for \(f(x) = x \cos(x)\), we multiply each term of the Maclaurin series for \(\cos(x)\) by \(x\): \[x \cos(x) = x \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n)!}.\]

Step 3: Write the Resulting Series

The resulting series for \(x \cos(x)\) is: \[x \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n)!}.\] This series sums over all terms where the exponents of \(x\) are odd, following the multiplication by \(x\).

Key Concepts

Cosine Function

The cosine function, represented as \(\cos(x)\), is a fundamental trigonometric function. It is important in both geometry and mathematics. The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. The function repeats its values every \(2\pi\), making it periodic.
Its graph is a smooth and continuous wave, oscillating between -1 and 1. This periodic behavior is essential in many fields, such as physics and engineering.
Understanding this function's behavior helps in series expansions and other calculus applications.

Series Expansion

Series expansion is a method of expressing functions as a sum of simpler terms. One common type is the Maclaurin series, a special case of the Taylor series centered at 0.
The goal is to approximate complex functions with a series of well-known polynomial terms. This approximation is made progressively better by increasing the number of terms. Series expansion is widely used in solving differential equations and approximating functions.

Power Series

A power series is an infinite sum of the form \(\sum_{n=0}^{\infty} a_n x^n\). In power series, each term is a power of \(x\) multiplied by a coefficient \(a_n\).
Power series are fundamental in representing functions analytically and are used in calculus to approximate functions over a certain interval.
Key aspects of power series:

• They converge within a radius from the center of expansion.
• They can represent functions precisely within their interval of convergence.
• Calculus operations like differentiation and integration can be performed term by term.

Calculus

Calculus is a branch of mathematics that studies continuous change. It includes differentiation and integration, fundamental operations that are used for a wide range of applications.
In Taylor and Maclaurin series, calculus is used to explore function behaviors and create approximations. This is particularly useful when exact values are difficult to compute.
With differentiation, we find the slope or rate of change of functions. Integration, on the other hand, is used to find areas under curves or total accumulation. These tools are crucial for deriving series like the one for \(x \cos(x)\).