Problem 50 - Calculus Early Transcendentals

Question

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\tan ^{-1}\left(\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

Answer: The first four nonzero terms of the Maclaurin series for the inverse tangent function evaluated at x=1/2 are: 1/2, -1/24, 1/160, and -1/896. The approximation using these terms is given by the sum: 1/2 - 1/24 + 1/160 - 1/896 ≈ 0.4647.

Step 1: Identify the Maclaurin series for the inverse tangent function

The Maclaurin series for the inverse tangent function, which is a specific case of the Taylor series when evaluated at \(x=0\), is given by: $$\tan^{-1}(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$

Step 2: Substitute the given value

We need to find the first four nonzero terms of the series for \(\tan^{-1}\left(\frac{1}{2}\right)\). To do this, simply substitute the given value, \(x = \frac{1}{2}\), into the Maclaurin series and find the first four nonzero terms. $$\tan^{-1}\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right) - \frac{\left(\frac{1}{2}\right)^3}{3} + \frac{\left(\frac{1}{2}\right)^5}{5} - \frac{\left(\frac{1}{2}\right)^7}{7} + \cdots$$

Step 3: Simplify each term

Now, simplify each term to obtain the first four nonzero terms of the infinite series: $$\tan^{-1}\left(\frac{1}{2}\right) = \frac{1}{2} - \frac{1}{2^3\cdot3} + \frac{1}{2^5\cdot5} - \frac{1}{2^7\cdot7} + \cdots$$ $$\tan^{-1}\left(\frac{1}{2}\right) = \frac{1}{2} - \frac{1}{24} + \frac{1}{160} - \frac{1}{896} + \cdots$$ The first four nonzero terms of the infinite series for the given value are: \(\frac{1}{2}\), \(-\frac{1}{24}\), \(\frac{1}{160}\), and \(-\frac{1}{896}\). The sum of these terms will give an approximation of \(\tan^{-1}\left(\frac{1}{2}\right)\).

Key Concepts

Maclaurin Series

The Maclaurin Series is a pivotal concept in calculus used to represent functions as infinite power series. It's a specific type of Taylor Series evaluated at zero. For a function \(f(x)\), the Maclaurin series is given by:
\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots\]
This series allows us to express complicated functions as a sum of its derivatives at a single point. It proves especially useful when dealing with functions like polynomials, sines, cosines, and exponentials, allowing these functions to be approximated to any desired degree of accuracy. Understanding how to construct and manipulate these series is crucial for solving complex calculus problems efficiently.

Inverse Trigonometric Functions

Inverse Trigonometric Functions, such as \( \tan^{-1}(x) \), are functions that reverse the effect of the standard trigonometric functions. When dealing with these functions, series like the Maclaurin series help in calculating values that are otherwise difficult to find using simple computations.
The power series provided by these expansions can approximate the inverse trigonometric functions at points where direct computation is hard. Inverse tangent, specifically, has a Maclaurin series:

• \( \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \)

The capability to write functions in terms of series is incredibly helpful for approximations needed in analysis or when working with computer calculations and simulations.

Series Approximation

Series Approximation is a method that provides a way to estimate the value of a function using a finite number of terms from its corresponding infinite series. In practical terms, we often use the first few terms because they offer a balance between simplicity and accuracy. This is particularly useful in engineering and computer science, where a quick estimation is beneficial.
For example, in finding \(\tan^{-1}(\frac{1}{2})\), using the first four nonzero terms from the Maclaurin series gives a useful approximation:

• \( \frac{1}{2} - \frac{1}{24} + \frac{1}{160} - \frac{1}{896} \)

Each step along the series provides an improved estimation, allowing us to get remarkably close to the actual value without needing to compute the entire infinite series.

Infinite Series

An Infinite Series is an endless sequence of numbers that, when summed, converge towards a specific value or diverge to infinity. It is a fundamental concept in mathematics that finds applications in a diverse range of fields, such as physics, finance, and computer algorithms.
In calculus, infinite series often denote the sum of infinite terms from a sequence. When truncated, they form the basis for many approximation methods. The series used are often power series like the Maclaurin or Laurant series, which can represent a vast array of functions accurately when enough terms are considered.

• The infinite series for \( \tan^{-1}(x) \) is one such example, where terms alternate in sign and decrease in magnitude rapidly.

Understanding infinite series is crucial to tackling problems in real-world scenarios where precision and accuracy are required over straightforward raw calculations.