Problem 55 - Thomas Calculus

Question

According to the Alternating Series Estimation Theorem, how many terms of the Taylor series for tan \(^{-1} 1\) would you have to add to be sure of finding \(\pi / 4\) with an error of magnitude less than \(10^{-3} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Add 500 terms to ensure the error is less than \(10^{-3}\).

Step 1: Identify the Alternating Series

The function \( \tan^{-1}(x) \) can be expressed as a Taylor series: \( \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \). For \( x = 1 \), this series becomes \( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \), which is an alternating series.

Step 2: State the Error Formula

According to the Alternating Series Estimation Theorem, the error after \( n \) terms of an alternating series is less than the absolute value of the next term: \( | R_n | < |a_{n+1}| \). The error must be less than \( 10^{-3} \).

Step 3: Calculate Successive Terms

List the absolute values of successive terms for \( x = 1 \): \( 1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \cdots \). Each term is \( \frac{1}{2n+1} \) where \( n \) denotes the term's position.

Step 4: Determine Minimum Terms Required

We need \( \frac{1}{2n+1} < 10^{-3} \). Solving \( 2n+1 > 1000 \), we find \( n > 499.5 \). So, \( n = 500 \) is needed to satisfy this condition.

Key Concepts

Taylor series

A Taylor series is a way to represent a function as an infinite sum of its derivatives evaluated at a single point. This series expansion allows complex functions to be approximated by simpler polynomial functions, which can be easier to work with for calculations and estimations.
Especially useful when dealing with infinity series, Taylor series are often utilized in calculus and mathematical analysis.
Some key features include:

• The Taylor series of a function will converge to the actual function value under certain conditions—especially if the function is infinitely differentiable around the point being evaluated.
• It is expressed in the form: \[ f(x) = \sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n \]
• This series can approximate functions around a specific point “a”, making it especially relevant in engineering, physics, and quantitative sciences.

In our specific problem, the Taylor series is used to approximate the value of the arctan function, one of the inverse trigonometric functions.

error estimation

Error estimation is crucial in determining the accuracy of approximations provided by series expansions, such as Taylor series. It helps us understand how close an approximation is to the exact value of the function being evaluated.
The Alternating Series Estimation Theorem, in particular, provides one of the simplest ways to estimate this error in specific types of series.
The theorem states:

• In an alternating series, the error of truncating the series (i.e., stopping after a certain number of terms) is less than the absolute value of the first omitted term.
• This error estimation is particularly useful because it gives a maximum bound on the error size, making it easier to control the accuracy of our approximations.
• In our task, we want the error to be less than 0.001. This condition helps determine the necessary number of terms to achieve the desired precision.

So, utilizing the error estimation principle ensures that computational results stay within acceptable error margins, maintaining reliability in various practical scenarios.

arctan function

The arctan function, or the inverse tangent function, is a fundamental trigonometric function often denoted as \( \tan^{-1}(x) \). It provides the angle whose tangent is the given number.
This function spans a primary range from \( -\pi/2 \) to \( \pi/2 \), making it essential in various mathematical applications where angle determination is needed.

• The arctan function can be represented by an infinite series when \( |x| \leq 1 \), offering a method for approximating its value using the Taylor series: \( \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots \)
• This representation is particularly powerful for small values of \( x \), where direct computation might be difficult but polynomial approximations are feasible.
• In our exercise, evaluating \( \tan^{-1}(1) \) effectively results in estimating the value \( \pi/4 \), underlining the arctan function's link to key mathematical constants.

The arctan function's utilization in the problem highlights its significance in both theoretical and practical applications, bridging complex trigonometric relationships with real-world computations.