Problem 14 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\sin x-x+\frac{x^{3}}{3 !}$$

Step-by-Step Solution

Answer

The Taylor series is \(\frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\).

Step 1: Understand the Problem

We need to find the Taylor series at \(x=0\) for the given function \(\sin x - x + \frac{x^3}{3!}\). A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point.

Step 2: Recall Taylor Series Form of Sine Function

The Taylor series for \(\sin x\) about \(x=0\) is given by \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\). We will use this series in our calculations.

Step 3: Substitute Taylor Series into Function

Substitute the series for \(\sin x\) into the function: \[ \sin x - x + \frac{x^3}{3!} = \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \right) - x + \frac{x^3}{3!}. \]

Step 4: Combine Like Terms

Simplify by combining like terms. \[ \sin x - x + \frac{x^3}{3!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots - x + \frac{x^3}{3!} = \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \].

Step 5: Conclusion

The reduced Taylor series at \(x=0\) for the function \(\sin x - x + \frac{x^3}{3!}\) simplifies to \(\frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \).

Key Concepts

Power Series

A power series is a series of the form:

• \[ a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \]

It features powers of a variable, typically denoted as "x," combined with coefficients. This series can be finite or infinite, with the latter having special importance in calculus as it closely approximates complex functions.

One powerful use is in creating the Taylor series, which approximates functions at a specific point using their derivatives. This concept shines in making highly accurate function approximations using only polynomial terms, especially around the Central point of expansion (often 0).

Power series are valuable in various fields, including physics and engineering, making it easier to work with complex calculations.

Sine Function

The sine function (\( \sin x \)) is a periodic function representing the y-coordinate of a point on the unit circle, given an angle. It is widely known for its wavelike pattern, oscillating between -1 and 1.

• Its period is \(2\pi\), meaning it repeats every \(2\pi\) radians.
• The sine function is odd, satisfying \(\sin(-x) = -\sin(x)\).

When expanded as a Taylor series, the sine function becomes an infinite series:

• \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \]

This expansion allows for accurate calculations of sine for small values of \(x\). It showcases the elegance of using series to simplify trigonometric functions.

Derivatives

Derivatives play a crucial role in calculus, providing information on the rate of change of functions. For any function \(f(x)\), its derivative \(f'(x)\) represents how the function changes with a slight variation in \(x\).

• First derivative: Measures instantaneous rate of change, giving rise to tangent lines in graphs.
• Second derivative: Offers insights into concavity of functions and finds applications in acceleration concepts in physics.

The Taylor series utilizes derivatives extensively. Each term in a Taylor series is derived from a derivative of the function at a specific point divided by the factorial of the term's degree.

For instance, the sine function's Taylor series involves alternating derivatives at \(x=0\), highlighting the significance of derivatives in defining accurate polynomial approximations.

Infinite Series

An infinite series is the summation of infinitely many terms. Starting from a basic sequence, we continuously add the sequence's terms. Mathematically, it is willing to look like:

• \[ a_1 + a_2 + a_3 + \cdots \]

For an infinite series to make sense or "converge," the summation must approach a specific value. This concept is deeply entrenched in calculus, allowing the breakdown of complex functions into simpler series forms.

Understanding and using infinite series is essential in Taylor expansions, as Taylor series are primarily infinite series used to expand functions around a point. By converging to the function’s value, these series enable precise approximations suitable for practical applications.