Problem 10 - A Course in Mathematical Methods for Physicists

Question

Consider Gregory's expansion $$ \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\cdots=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2 k+1} x^{2 k+1} $$ a. Derive Gregory's expansion using the definition $$ \tan ^{-1} x=\int_{0}^{x} \frac{d t}{1+t^{2}} $$ expanding the integrand in a Maclaurin series, and integrating the resulting series term by term. b. From this result, derive Gregory's series for \(\pi\) by inserting an appropriate value for \(x\) in the series expansion for \(\tan ^{-1} x\).

Step-by-Step Solution

Answer

The Gregory's series for \( \tan^{-1} x \) is \( \tan^{-1}x = \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} x^{2k+1} \). And the derived Gregory's series for \( \pi \) is \( \pi = \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} * 4 \).

Step 1: Expanding the Integrand

Start by expanding \( \frac{1}{1 + t^2} \) in terms of a Maclaurin series. This becomes \( 1 - t^2 + t^4 - t^6 + \cdots \) or \( \sum_{k=0}^{\infty} {(-1)^k t^{2k}} \).

Step 2: Integration the series term by term

Now you need to integrate this series term by term from 0 to x. The result is \( x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \) or \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} {x^{2k+1}} \). This proves the first part of Gregory's expansion.

Step 3: Derivation of Gregory's series for \( \pi \)

Now to derive Gregory's series for \( \pi \), an appropriate value of \( x \) needs to be inserted in the series. The arctangent function has a value of \( \pi/4 \) for \( x = 1 \). So, putting \( x = 1 \) in the series, we have \( \pi / 4 = 1 - 1/3 + 1/5 - 1/7 + \cdots \) or \( \pi = 4 * (1 - 1/3 + 1/5 - 1/7 + \cdots) = \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} * 4 \). This is the required Gregory's series for \( \pi \).

Key Concepts

Maclaurin series

A Maclaurin series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Specifically, for a function that is infinitely differentiable at the point zero, the Maclaurin series is a type of Taylor series that expands around zero. It is expressed as:
\[ f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \]
In simpler terms, the Maclaurin series takes the complex behavior of functions and breaks them down into polynomials, which are easier to understand and work with, especially when approximating the values of functions near the point of expansion.

Integration of series

The process of integrating a series involves taking the integral of each term within the series individually. For a series represented by \( \sum_{k=0}^{fty} a_k(x) \), where \( a_k(x) \) are functions of \( x \), the integral is found by calculating \( \int \sum_{k=0}^{fty} a_k(x) dx \), which is equal to \( \sum_{k=0}^{fty} \int a_k(x) dx \). It's essential to ensure the series is uniformly convergent within the interval of integration to legally switch the summation and integration operations.
Integrating series term by term is a powerful tool as it simplifies complex operations into manageable calculations, making it possible to find explicit expressions for functions that are otherwise difficult to integrate directly.

Gregory's series for pi

Gregory's series for \( \pi \) is a special case of Gregory's expansion, which derives from the Maclaurin series of the arctangent function. By setting \( x = 1 \) in the series for the arctangent function, we obtain a representation of \( \pi \) as an infinite series:
\[ \pi = 4\left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right) = 4 \sum_{k=0}^{fty} \frac{(-1)^k}{2k+1} \]
This equation expresses \( \pi \) as an alternating series, which converges quite slowly, but nonetheless, provides an exact representation of \( \pi \) and a method for approximating it to any desired level of accuracy. Gregory's series has historical significance and is one of the earliest examples of series expressions for transcendental numbers.

Arctangent function

The arctangent function, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is the inverse of the tangent function. It returns the angle whose tangent is \( x \), and it is defined for all real numbers \( x \). An important property of the arctangent function is that \( \arctan(1) \) equals \( \pi/4 \), a relationship that is crucial in deriving Gregory's series for \( \pi \). The arctangent function can be expressed as an infinite series, which was shown by Gregory's expansion:
\[ \tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots = \sum_{k=0}^{fty} \frac{(-1)^k}{2k+1} x^{2k+1} \]
The series is the result of integrating the geometric series of \( \frac{1}{1+t^2} \), and it converges for all \( |x| \leq 1 \). It's a central component in calculus, especially when working with trigonometric integrals and series expansions.