Problem 15 - Calculus

Question

Chapter 10 gives an infinite series for \(\sin x\) : $$ \sin x=\frac{x}{1}-\frac{x^{3}}{3 \cdot 2 \cdot 1}+\frac{x^{5}}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}-\cdots $$ From the derivative find the series for \(\cos x\). Then take its derivative to get back to \(-\sin x\)

Step-by-Step Solution

Answer

The series for \(\cos x\) is \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\). The derivative of this series gives \(-\sin x\).

Step 1: Understand the Given Series for \(\sin x\)

The series for \(\sin x\) is given by a power series expansion: \(\sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\). This is an alternating series with odd powers of \(x\). Our goal is to find the series for \(\cos x\) starting from this.

Step 2: Find the Derivative of \(\sin x\)

To find the series for \(\cos x\), take the derivative of the given series for \(\sin x\).The derivative of \(\frac{x^n}{n!}\) is \(\frac{d}{dx}\left(\frac{x^n}{n!}\right) = \frac{nx^{n-1}}{n!} = \frac{x^{n-1}}{(n-1)!}\).Thus, the derivative of the series is:\[ \cos x = \frac{d}{dx}\left(\sin x\right) = \frac{1}{1!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \]

Step 3: Verify the Series for \(\cos x\)

We have derived the series \(\cos x = \frac{1}{1!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\), which is correct for \(\cos x\). Confirm that it is composed of terms with even powers of \(x\).

Step 4: Take Derivative of \(\cos x\) to Recover \(-\sin x\)

Now, take the derivative of the series for \(\cos x\) to retrieve the series for \(-\sin x\):\(\frac{d}{dx}\left(\cos x\right) = \frac{d}{dx}\left(\frac{1}{1!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right)\)The derivative of this series is:\[ -\sin x = 0 - \frac{2x}{2!} + \frac{4x^3}{4!} - \cdots \ = -\left(\frac{x}{1!} - \frac{x^3}{3!} + \cdots\right) \]

Step 5: Confirm Recovery of \(-\sin x\)

The derivative produces the series \(-\left(\frac{x}{1!} - \frac{x^3}{3!} + \cdots\right)\), confirming the recovery of \(-\sin x\). Each term has the sign inverted and matches the terms of the original \(\sin x\) series which were negated.

Key Concepts

Power Series

A power series is a way of expressing a function as an infinite sum of terms. Each term involves a power of the variable. The general form of a power series centered at zero is given by: \[ f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + \cdots = \sum_{n=0}^{\infty} c_n x^n \]

• Convergence: The power series can be used whenever it converges, which depends on the specific series and its radius of convergence.
• In Calculus: Power series are crucial because they allow complicated functions to be expressed in simpler polynomial forms which are easy to differentiate and integrate.

When applied to functions like \( \sin x \) and \( \cos x \), power series provide a way to calculate these trigonometric functions numerically for any value of \( x \). These series are especially handy when dealing with wave equations and oscillations in physics.

sin x

The function \( \sin x \) can be expressed as a power series. As provided: \[ \sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \] This alternating series uses only odd powers and factorials, with alternating positive and negative signs.

• Calculation Insight: Each term's contribution becomes smaller due to the increasing factorial in the denominator.
• Practical Use: Such series is useful when calculating values of \( \sin x \) without using a calculator or understanding its behavior for different values of \( x \).

Notably, Taylor series allow approximation of \( \sin x \) for small angles by truncating the series to a few terms.

cos x

The function \( \cos x \) is similarly expressed through its power series. Derived from the derivative of the \( \sin x \) series: \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \] This series involves even powers of \( x \) and follows a similar alternating pattern.

• Derivative Link: The series for \( \cos x \) results from differentiating the \( \sin x \) series, showcasing how trigonometric functions are interrelated.
• Understanding Behavior: This series illustrates \( \cos x \) behavior which oscillates between -1 and 1 with zeroes at \( \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \), etc.

The power series for \( \cos x \) provides another powerful tool in analysis and practical computation of trigonometric functions.

Derivatives

Derivatives are a fundamental concept in calculus, giving information about the rate of change. The derivative of a function provides the slope of the function curve at a specific point.

• Chain of Changes: Taking the derivative of series like \( \sin x \) and \( \cos x \) shows how a small change in input affects the function output and interrelates these trigonometric functions.
• From \( \sin x \) to \( \cos x \): Differentiating \( \sin x \) leads to \( \cos x \).
• Back to \( -\sin x \): Differentiating \( \cos x \) again gives \( -\sin x \), exemplifying cyclical interchange of these functions.

In mathematical analysis, derivatives unveil connections between various functions, allowing deeper investigation into their properties and applications in physics and engineering.