Problem 56 - Thomas Calculus

Question

Show that the Taylor series for \(f(x)=\tan ^{-1} x\) diverges for \(|x|>1 .\)

Step-by-Step Solution

Answer

The Taylor series for \(f(x)=\tan^{-1}x\) diverges for \(|x|>1\).

Step 1: Understanding the Taylor Series

The Taylor series for a function centered at zero is given by \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]For the function \( f(x) = \tan^{-1}x \), we need to compute the derivatives at \( x=0 \).

Step 2: Calculating Derivatives

The first few derivatives of \( f(x) = \tan^{-1}x \) are as follows:1. \( f'(x) = \frac{1}{1+x^2} \), so \( f'(0) = 1 \).2. \( f''(x) = -\frac{2x}{(1+x^2)^2} \), so \( f''(0) = 0 \).3. \( f'''(x) = -\frac{2(1-x^2)}{(1+x^2)^3} \), so \( f'''(0) = -2 \).The pattern for the derivatives at \( x=0 \) can be used to form the series.

Step 3: Forming the Taylor Series

Substitute the derivative values obtained at \( x=0 \) into the Taylor series formula:\[ \tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \]This series resembles a known mathematical series called the power series.

Step 4: Identifying the Power Series

The derived series is the power series obtained for \( \tan^{-1}x \), which is a known alternating series:\[ \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \]It converges for \(|x| \leq 1\), known behavior for these types of series.

Step 5: Analyzing Convergence

For a power series \( \sum_{n=0}^{\infty} a_n x^n \), convergence depends on \(|x|\) relative to the radius of convergence. Here, \( a_n = \frac{(-1)^n}{2n+1} \) and convergence radius is determined to be 1. The interval \(|x| \leq 1\) supports convergence.

Step 6: Demonstrating Divergence Beyond the Radius

If \(|x| > 1\), the series \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \) diverges since it exceeds the radius of convergence calculated to be 1. Thus, the series diverges when \(|x| > 1\). This is a property derived from analyzing the terms behavior as \( n \to \infty \).

Key Concepts

Radius of Convergence

The radius of convergence plays a crucial role in determining where a Taylor series can be trusted to accurately represent its function. In simple terms, the radius of convergence is a boundary around the center of the series (usually at zero) within which the series converges to the actual value of the function. Beyond this boundary, the series fails to converge, or in other words, it diverges.
To find the radius of convergence for a given power series, we apply the ratio test or the root test. For example, for the Taylor series of the arctangent function, \[\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1},\]the radius of convergence can be determined using such techniques.
The vital point to remember is that this series will only converge when \(|x| \leq 1\). As soon as \(|x|\) becomes greater than 1, it steps outside the radius of convergence, causing the series to diverge.
Understanding the radius of convergence helps us in knowing the limits of the Taylor series and ensuring its correct application.

Power Series

A power series is a series of the form \[\sum_{n=0}^{\infty} a_n (x-c)^n,\]where \(a_n\) are coefficients and \(c\) is the center of the series. Power series provide a way to express many functions in terms of an infinite polynomial, which is especially useful for approximation.
When dealing with power series, we concern ourselves with where the series converges, which is controlled by the radius of convergence. A power series converges within its radius and possibly at the endpoints. It gives us a versatile tool to handle functions, allowing them to be expressed in simpler polynomial terms.
The Taylor series derived from the arctangent function can be thought of as a specific form of power series. This ties into how Taylor series are a subset of power series, specializing in the function's expansion around a single point. By analyzing a power series's terms and behaviors, it guides us towards understanding its convergence, assisting in both theoretical and practical endeavors.

Arctangent Function

The arctangent function, denoted as \(\tan^{-1}x\) or \(\arctan(x)\), is an inverse trigonometric function that outputs an angle whose tangent is \(x\). This function is important in many areas of mathematics, including calculus, where it is essential for tasks such as integration and solving triangles.
The Taylor series for the arctangent function is an important representation, providing an infinite series formula:\[\tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\]This series is known as an alternating series, which alternates signs between each term. Alternating series can behave differently depending on the value of \(x\), primarily converging for \(|x| \leq 1\).
This behavior of the arctangent function's series highlights how enormously powerful Taylor series can be, offering us a way to extend function concepts across wide domains. However, it also underlines the necessity to understand convergence, showing that improper use can lead to inaccurate results.