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 \(x\) . a. Find the first six terms of a series for cos \(x\) . For what values of \(x\) should the series converge? b. By replacing \(x\) by 2\(x\) in the series for \(\sin x,\) find a series that converges to \(\sin 2 x\) for all \(x .\) c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for 2 \(\sin x \cos x\) . Compare your answer with the answer in part (b).

Step-by-Step Solution

Answer

The Cosine series: \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\). The series converges for all \(x\). For \(\sin(2x)\), replace \(x\) by \(2x\) in \(\sin x\). Both steps (b) and (c) result in the same series for \(\sin(2x)\).

Step 1: Expand Series for cos x

The Maclaurin series for \( \cos x \) is given by:\[\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \cdots\]These are the first six terms. The series converges for all \( x \), similar to \( \sin x \).

Step 2: Apply Transformation for sin(2x)

Substitute \( 2x \) for \( x \) in the series for \( \sin x \):\[\sin(2x) = (2x) - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \cdots\]Simplifying, this becomes:\[\sin(2x) = 2x - \frac{8x^3}{6} + \frac{32x^5}{120} - \cdots\]This series converges for all \( x \).

Step 3: Use Series Multiplication for 2 sin x cos x

Using the series for \( x - \frac{x^3}{3!} + \cdots \) for \( \sin x \) and \( 1 - \frac{x^2}{2!} + \cdots \) for \( \cos x \), calculate:\[2 \sin x \cos x = 2(x - \frac{x^3}{6} + \cdots)(1 - \frac{x^2}{2} + \cdots)\]Multiply the series together and simplify terms up to the fifth degree. The first six terms after simplification will match the series for \( \sin(2x) \) obtained earlier in Step 2.

Key Concepts

Maclaurin series

The Maclaurin series is a specific type of Taylor series centered at zero. It is a way of expressing functions as an infinite sum of terms calculated from the derivatives of a function at a single point, zero. This is particularly useful in approximating functions that are otherwise difficult to work with. For trigonometric functions like sine and cosine, the Maclaurin series gives expressions that allow us to evaluate these functions without a calculator for small values of x.
For example, the Maclaurin series for \( \sin x \) is:

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

Similarly, for \( \cos x \) it is:

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

These series allow for precise approximations based on the number of terms you include.

convergence of series

When we talk about the convergence of a series, we are referring to whether the sum of its infinite terms approaches a specific value as more terms are added. For example, the Maclaurin series for \( \sin x \) and \( \cos x \) both converge absolutely for all real numbers \( x \). This means that no matter what value you plug in for \( x \), the series will approach the actual sine or cosine value.
The convergence is ensured because the terms added to form these series decrease in magnitude factorially, meaning each term rapidly becomes smaller, leading to a stable sum. This mathematical property is crucial for practical calculations, ensuring that as you add more terms, your approximation becomes closer to the true value.

trigonometric series

Trigonometric series are infinite series derived from the trigonometric functions \( \sin \) and \( \cos \). These series allow us to represent these periodic functions in terms of algebraic expressions that can be differentiated, integrated, or manipulated more easily than the original functions.
The significance of trigonometric series becomes evident in various applications, including solving differential equations, signal processing, and Fourier analysis. For example, you can use series expansions of \( \sin x \) and \( \cos x \) to find solutions for wave-like phenomena without delving into complex computations manually.

series expansion

Series expansion is a technique used to express a complex function as an infinite sum of simpler terms. This method is particularly beneficial in mathematics for simplifying calculations and approximations. It involves writing the function as a power series, often derived from the Taylor series at a specified point.
A series expansion allows for an easier approach to working with non-linear functions. For instance, expanding \( \sin x \) and \( \cos x \) in terms of power series lets us solve various mathematical problems in areas like physics and engineering more efficiently. By using a series, you can also compute unknown values by substituting into the series and truncating it for a practical result.

sin and cos identities

The identities related to \( \sin x \) and \( \cos x \) are foundational tools in trigonometry. These identities help simplify expressions and solve equations involving trigonometric functions. Numerous identities connect \( \sin \) and \( \cos \) to each other and to other trigonometric functions.
Among the most used are the double-angle identities, such as:

• \( \sin(2x) = 2 \sin x \cos x \)
• \( \cos(2x) = \cos^2 x - \sin^2 x \)

These identities underline how the products and powers of \( \sin \) and \( \cos \) can be transformed into single trigonometric functions, making it easier to handle complex trigonometric expressions or integrate and differentiate them smoothly. Understanding and using these identities can greatly enhance your problem-solving skills in mathematics.