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

501 terms are needed to ensure the error is less than 0.001.

Step 1: Introduction to the Problem

The problem is to determine how many terms of the Taylor series for \( \tan^{-1}(1) \) we need to use to approximate \( \frac{\pi}{4} \) with an error less than \( 10^{-3} \). This Taylor series is given as \( \sum_{n=0}^{\infty} (-1)^n \frac{1}{2n+1} \). According to the Alternating Series Estimation Theorem, the error is bounded by the absolute value of the first omitted term.

Step 2: Formulate the Error Condition

To satisfy the Alternating Series Estimation Theorem, we require \( \left| a_{n+1} \right| < 10^{-3} \), where \( a_{n+1} = \frac{1}{2n+1} \). Hence, we need \( \frac{1}{2n+1} < 10^{-3} \).

Step 3: Inequality Calculation

Solving the inequality \( \frac{1}{2n+1} < 10^{-3} \), we multiply both sides by \( 2n+1 \) and divide by \( 10^{-3} \): \( 1 < 10^{-3} \times (2n+1) \).

Step 4: Simplify the Inequality

This simplifies to \( 1 < 0.001(2n+1) \), leading to \( 1000 < 2n+1 \). Thus, after subtracting 1 from both sides, \( 999 < 2n \).

Step 5: Solve for n

Divide both sides by 2 to solve for \( n \): \( n > 499.5 \). Therefore, the smallest integer \(n\) that satisfies this inequality is \( n = 500 \).

Step 6: Conclusion

We require 501 terms of the Taylor series (counting from 0 to 500) to ensure the error is less than \( 10^{-3} \).

Key Concepts

Alternating Series Estimation Theorem

The Alternating Series Estimation Theorem is a very handy tool when working with alternating series like the one in our exercise. This theorem helps us estimate the error we make when approximating the sum of an infinite series with a finite number of terms. An alternating series is simply a series where consecutive terms alternate in sign. In mathematical terms, it can be written as \(a_1 - a_2 + a_3 - a_4 + \ldots \), where all \(a_n\) are positive. The beauty of the theorem is that it gives us a straightforward way to know how accurate our approximation is.

When applying the Alternating Series Estimation Theorem, if the terms \(a_n\) in the series are decreasing in absolute value and converge to zero, then the error made by using the first \(n\) terms as an approximation is no larger than the absolute value of the first omitted term. This gives us a practical way to control the magnitude of our estimation error. For instance, in our solved exercise, calculating the error involved knowing when \(\left| a_{n+1} \right| < 10^{-3}\), ensuring our solution falls within the defined error bound.

Error Estimation

Error estimation is crucial in mathematical series and approximations for ensuring our calculations are close enough to the true value. Particularly in series like Taylor series, error estimation tells us how far off our sum is from the actual infinite series sum after using only a finite number of terms. In the scenario of the exercise, we're interested in knowing the smallest number of terms needed to approximate \(\frac{\pi}{4}\) with an error less than \(10^{-3}\).

To achieve a specific precision, we must scrutinize the terms of the series. By using the criterion \(\left| a_{n+1} \right| < \epsilon\), where \(\epsilon\) represents the desired error margin (here, \(10^{-3}\)), we ensure we don't overstep the allowable error. For the Taylor series of \(\tan^{-1}(1)\), the terms decrease in size; therefore, the first term omitted after reaching this error threshold provides a quick measure of our maximum possible error.

Infinite Series

An infinite series is the sum of an endless sequence of numbers, which provides a way to describe functions in terms of simpler components. Taylor series, a type of infinite series, allow us to represent complex functions using polynomials. These polynomials can be extended indefinitely, hence the name "infinite". The key to working with infinite series is understanding that though they have an infinite number of terms, practical operations involve only a finite number of terms for estimation purposes.

The exercise involves the Taylor series expansion of \(\tan^{-1}(1)\), representing this angle in terms of an ever-growing sum. Infinite series like this one continuously converge toward a more accurate representation of a value as more terms are added. Yet, they also necessitate careful consideration of convergence and the error associated with truncating the series. Convergence points towards the series reaching a specific value, and alternatively, divergence shows if a series can't settle on a value. Knowing when and how much of the series to use forms the basis of many applied math problems, as demonstrated by the step-by-step solution in the exercise.