Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Did you ever wonder how people find the decimal expansion of \(\pi\) to a large number of places? One method depends on the following identity (see Problem 34 of Section 1.9). $$ \pi=16 \tan ^{-1}\left(\frac{1}{5}\right)-4 \tan ^{-1}\left(\frac{1}{239}\right) $$ Find the first 6 digits of \(\pi\) using this identity and the series for \(\tan ^{-1} x\). (You will need terms through \(x^{9} / 9\) for \(\tan ^{-1}\left(\frac{1}{5}\right)\), but only the first term for \(\left.\tan ^{-1}(1 / 239) .\right)\) In 1706, John Machin used this method to calculate the first 100 digits of \(\pi\), while in 1973 , Jean Guilloud and Martine Bouyer found the first 1 million digits using the related identity $$ \pi=48 \tan ^{-1}\left(\frac{1}{18}\right)+32 \tan ^{-1}\left(\frac{1}{57}\right)-20 \tan ^{-1}\left(\frac{1}{239}\right) $$ In \(1983, \pi\) was calculated to over 16 million digits by a somewha different method. Of course, computers were used in these recen calculations.

Step-by-Step Solution

Answer

The first 6 digits of \(\pi\) are 3.14159.

Step 1: Understand the Series for arctan

The series for the arctangent function is given by:\[\tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots\]We will use this series to approximate \(\tan^{-1}\left(\frac{1}{5}\right)\) and \(\tan^{-1}\left(\frac{1}{239}\right)\).

Step 2: Compute \(\tan^{-1}\left(\frac{1}{5}\right)\)

Substitute \(x = \frac{1}{5}\) in the arctan series:\[\tan^{-1}\left(\frac{1}{5}\right) = \frac{1}{5} - \frac{1}{3 \times 5^3} + \frac{1}{5 \times 5^5} - \frac{1}{7 \times 5^7} + \frac{1}{9 \times 5^9}\]Simplifying each term:\[= 0.2 - 0.0013333 + 0.000064 + 0.0000022857 - 0.000000064\]Add these terms together to get an approximate value of \(\tan^{-1}\left(\frac{1}{5}\right)\).

Step 3: Compute \(\tan^{-1}\left(\frac{1}{239}\right)\)

Since we only need the first term:\[\tan^{-1}\left(\frac{1}{239}\right) \approx \frac{1}{239} \approx 0.0041841\]Only the first term is needed as higher powers of \(\frac{1}{239}\) contribute insignificantly for six-digit precision.

Step 4: Combine to Find \(\pi\)

Substitute these approximations back into the identity:\[\pi \approx 16 \times 0.1973956 - 4 \times 0.0041841\]Calculate each part individually:\[16 \times 0.1973956 = 3.1583296\]\[4 \times 0.0041841 = 0.0167364\]Then:\[\pi \approx 3.1583296 - 0.0167364 = 3.1415932\]Rounding this result gives the first 6 digits: 3.14159.

Key Concepts

Arctangent Series

The arctangent series is a foundational concept in calculus, used to approximate the arctan function through an infinite series expansion. The series for \( an^{-1}(x)\) is represented as: \[ \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots \]This series allows mathematicians to approximate values for arctangent functions by using the first few terms of the series. For instance, in the calculation of \( an^{-1}\left(\frac{1}{5}\right)\), the series provides a way to find approximate decimal values by expanding each term based on powers of \(rac{1}{5}\). Similarly, for smaller values of \(x\), like \(rac{1}{239}\), only the first term is often needed, as subsequent terms become negligible. By switching values of \(x\) in the series, the formula adapts to compute different arctangent values.

Decimal Expansion

Decimal expansion refers to expressing numbers in their decimal form, which is particularly important for numbers like \(\pi\) that are irrational and cannot be exactly represented as fractions. Calculating the decimal expansion of \(\pi\) involves long sequences of calculations, often achievable through series and identities. By using identities involving arctangent functions, a large number of digits can be obtained. When using a series, each term contributes to extending the number of calculated decimal places. For example, by using a sequence up to \(x^9/9\) in the series, each term adds precision and extends the decimal expansion further.

Pi Calculation Methods

Over centuries, mathematicians have developed numerous methods to calculate \(\pi\) to ever-greater numbers of digits. Early methods relied on geometric approaches, while modern techniques use series expansions and computers. Among these is the arctangent series method, as featured in the exercise. By exploiting the identity \[ \pi = 16 \tan^{-1}\left(\frac{1}{5}\right) - 4 \tan^{-1}\left(\frac{1}{239}\right) \]Machin in 1706 calculated \(\pi\) to 100 digits using this formula. As computational power increased, more complex series and identities emerged, allowing millions of digits to be calculated. The series method remains a fundamental approach, augmented today by computer algorithms that optimize these calculations.

Historical Calculations of Pi

The quest to calculate \(\pi\) has a long and storied history, marked by the gradual increase in precision over centuries. Early estimations by ancient mathematicians such as Archimedes used geometric methods, like inscribing and circumscribing polygons. These early estimates formed the basis for more rigorous methods developed over time. In 1706, John Machin achieved a landmark with his arctangent identity, calculating \(\pi\) to 100 decimal places. This opened doors to the use of new identities and arithmetic innovations, significantly boosting precision. In 1973, Jean Guilloud and Martine Bouyer managed to calculate the first million digits using an extended arctangent identity.With the advent of computers, historical methods evolved. By the 1980s, computational techniques allowed \(\pi\) to be determined to millions of digits, illustrating how human ingenuity and technological advances have transformed mathematical challenges over time.