Problem 55 - Thomas Calculus

Question

The series $$ \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\frac{x^{11}}{11 !}+\cdots $$ converges to sin \(x\) for all \(x\) $$ \begin{array}{l}{\text { a. Find the first six terms of a series for cos } x \text { . For what values }} \\ {\text { of } x \text { should the series converge? }} \\ {\text { b. By replacing } x \text { by } 2 x \text { in the series for sin } x, \text { find a series that }} \\ {\text { converges to } \sin 2 x \text { for all } x \text { . }}\\\\{\text { c. Using the result in part (a) and series multiplication, calculate }} \\ {\text { the first six terms of a series for } 2 \sin x \text { cos } x \text { . Compare your }} \\\ {\text { answer with the answer in part (b). }}\end{array} $$

Step-by-Step Solution

Answer

The series for \(\cos x\) is 1 minus even powers of \(x\). It converges for all \(x\). Replacing \(x\) with \(2x\) in \(\sin x\) yields a series for \(\sin 2x\). \(2 \sin x \cos x\) matches \(\sin 2x\).

Step 1: Understanding the Problem

We need to find the first six terms of a series expansion for \(\cos x\), determine its convergence, transform the \(\sin x\) series to find \(\sin 2x\), and use series multiplication to derive \(2 \sin x \cos x\).

Step 1: Find the Series for \(\cos x\)

The Taylor series for \(\cos x\) around \(x = 0\) is: \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \cdots \] These are the first six non-zero terms of the series for \(\cos x\).

Step 2: Determine the Convergence of the \(\cos x\) Series

As a Taylor series centered at 0 based on powers of \(x\), it converges for all real numbers \(x\) because it's an even function with the entire domain of convergence.

Step 3: Replace \(x\) with \(2x\) in the Series for \(\sin x\)

Replace \(x\) with \(2x\): \[ \sin 2x = 2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \frac{(2x)^7}{7!} + \cdots \] Simplifying these terms gives: \[ \sin 2x = 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \cdots \]

Step 4: Compute \(2 \sin x \cos x\) Using Series Multiplication

Multiply the series for \(\sin x\) and \(\cos x\) term by term to find the first few terms of \(2 \sin x \cos x\). Start with \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\) and \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \). The first terms after multiplying with 2: \(2x - \frac{2x^3}{3} - \text{higher order terms}\).

Step 5: Compare the Answers for \(2\sin x \cos x\) and \(\sin 2x\)

Check if the first few terms of \(2\sin x \cos x\) found through multiplication match the simplified expansion of \(\sin 2x\). They should align, confirming the identity \(\sin 2x = 2 \sin x \cos x\).

Key Concepts

Series Expansion

A series expansion is a way to express a function as a sum of terms calculated from the values of its derivatives at a single point. In simple terms, it's like breaking down a function into an infinite series of simpler parts. This can make complex functions easier to understand and compute.
For example, the series expansion for \(\sin x\) is given by:

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

This series continues infinitely, and each additional term improves the approximation of \(\sin x\) for a given \(x\). Similarly, we can write the Taylor series of \(\cos x\) at \(x = 0\):

• \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\)

These expansions help us to approximate functions like trigonometric functions, which are often used in calculus and physics. Understanding series expansions is a key step in solving many mathematical problems, especially those involving calculus.

Convergence

Convergence refers to the idea that as you add more terms to a series, the result approaches a particular value, known as the limit. It's like getting closer and closer to a specific point as you add more steps.
In the context of a Taylor series, such as the series for \(\sin x\) and \(\cos x\), the series converges if the values it produces get increasingly accurate as more terms are added. For these trigonometric series, convergence means the series reliably estimates the original function across specified intervals.
For instance, both the \(\sin x\) and \(\cos x\) series converge for all real numbers, denoted by \(\mathbb{R}\). This convergence is guaranteed because they are well-behaved mathematical functions with infinite domains. Understanding when a series converges is crucial. It tells us if and where the series can be used to approximate a function accurately in calculus and other mathematical applications.

Trigonometric Series

Trigonometric series involve expansively expressing trigonometric functions, like \(\sin x\) and \(\cos x\), using infinite sums. This technique is rooted in calculus and is fundamental in fields that apply mathematics, such as engineering or physics.
The trigonometric series expansion for \(\sin x\) is expressed as:

• \(\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots\)

Similarly, the trigonometric series for \(\cos x\) is:

• \(\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots\)

These series are the backbone of Fourier analysis, which breaks down periodic functions into sums of simple sine and cosine waves.
The knowledge of trigonometric series aids in comprehending complex wave patterns or oscillations in various scientific and technical domains. This insight is essential when calculating precise waveforms and fluctuations in disciplines ranging from signal processing to quantum physics.