Problem 23 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x \tan ^{-1} x^{2}$$

Step-by-Step Solution

Answer

The series is \( x^3 - \frac{x^7}{3} + \frac{x^{11}}{5} - \frac{x^{15}}{7} + \cdots \).

Step 1: Recall the Taylor Series for the Arctangent Function

The function we want to work with is \( an^{-1}(x^2)\). To find its series, recall that the Taylor series for \( an^{-1}(x)\) around \(x = 0\) is given by: \[tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\]Replacing \(x\) with \(x^2\), we get:\[tan^{-1}(x^2) = x^2 - \frac{x^6}{3} + \frac{x^{10}}{5} - \frac{x^{14}}{7} + \cdots\]

Step 2: Multiply the Series by x

Our goal is to find the series representation for \(x an^{-1}(x^2)\). Multiply each term of the series \(\tan^{-1}(x^2)\) by \(x\):\[x an^{-1}(x^2) = x(x^2 - \frac{x^6}{3} + \frac{x^{10}}{5} - \frac{x^{14}}{7} + \cdots)\]This simplifies to:\[x^3 - \frac{x^7}{3} + \frac{x^{11}}{5} - \frac{x^{15}}{7} + \cdots\]

Step 3: Summarize the Taylor Series

The Taylor series for \(x \tan^{-1}(x^2)\) centered at \(x = 0\) is given by the infinite series:\[x an^{-1}(x^2) = x^3 - \frac{x^7}{3} + \frac{x^{11}}{5} - \frac{x^{15}}{7} + \cdots\]This series includes the dominant higher-order terms that arise from multiplying the given series by \(x\).

Key Concepts

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n(x-c)^n \), where \(a_n\) represents the coefficient of each term, \(x\) is the variable, and \(c\) is the center of the series. It's a way of expressing functions as infinite sums of terms involving powers of \(x-c\). The beauty of power series lies in their ability to represent complex functions using simple polynomials, which makes calculations and approximations more manageable.

• **Convergence**: A power series converges within a certain interval known as its radius of convergence. Within this interval, the series approaches a specific function as the number of terms increases.
• **Polynomial Approximation**: The more terms you include, the closer the series gets to the actual function within its convergence radius.

Power series are flexible, allowing transformations and manipulations, such as differentiation and integration, which helps when solving calculus problems.

Arctangent Function

The arctangent function, written as \( \tan^{-1}(x) \), is the inverse of the tangent function on a restricted domain. It produces an angle whose tangent is the given number. In calculus, it's significant due to its smooth, continuous curve that holds neat properties for series expansion.The Taylor series expansion of \( \tan^{-1}(x) \) is crucial for understanding how we can express this function as a power series:\[\tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\]

• **Inverse Nature**: It's valued between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), giving the principal value of the angle.
• **Applications**: The arctangent function is frequently used in trigonometry, calculus, and coordinate transformations.

Being able to express \( \tan^{-1}(x) \) in a series form makes it easier to calculate angles and solve complex integrals.

Series Expansion

Series expansion involves expressing a function in terms of the sum of simpler functions, often polynomials. The Taylor series is one of the most commonly used series expansions, representing functions as infinite sums of derivatives at a single point.To illustrate, consider the task at hand to find the Taylor series representation of \( x \tan^{-1}(x^2) \). This involves replacing \( x \) with \( x^2 \) in the series for \( \tan^{-1}(x) \) and multiplying each term by \( x \):\[x \tan^{-1}(x^2) = x(x^2 - \frac{x^6}{3} + \frac{x^{10}}{5} - \frac{x^{14}}{7} + \cdots)\]

• **Modification**: Adjusting the variable within the series and multiplying by another term allows us to derive new series specifically tailored to a given function.
• **Infinite Nature**: A complete series expansion extends infinitely, but practical use often involves truncating the series after a few terms for approximation.

Series expansions are powerful for approximating values of functions that do not have simple closed forms, offering deeper insights into their behavior at specified points.