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Chapter 1: Crises in Mathematics: Fourier's Series

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

The Arctangent Series

Wallis's Product

> Newton's Binomial Series

Ramanujan's Series

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts

2.3 Calculating Pi (continued)

Newton's Binomial Series

In 1665, Isaac Newton read Wallis's Arithmetica infinitorum in which he explains how to derive his product identity. This led Newton to an even more important discovery. He described the process in a letter to Gottfried Leibniz written on October 24, 1676.

The starting point was to both generalize and simplify Wallis's integral. Newton looked at When m is an even integer, we can use the binomial expansion to produce a polynomial in x:

What happens when m is an odd integer? Is it possible to interpolate between these polynomials? If it is, then we could let m = 1 and x = 1 and obtain an expression for /4.

Newton realized that the problem comes down to expanding as a polynomial in and then integrating each term. Could this be done when m is an odd integer?

Playing with the patterns that he discovered, he stumbled upon the fact that not only could he find an expansion for the binomial when the exponent is m/2, m odd, he could get the expansion with any exponent. Unless the exponent is a positive integer, the expansion is an infinite series.

In chapter 4, we will see when this expansion is also valid at x = 1 or –1.

Equipped with equation (2.3.7), we can approximate /4:

(2.3.8)

This series is an improvement, but Newton showed how to use his binomial series to do much better. He found that

(2.3.9)

Click here to explore its convergence to .

All of this work is fraught with potential problems. How do we know we are allowed to integrate the infinite summation by integrating each term? As we shall see later, integrating each term can lead to errors. Here it works.

Newton's discovery was more than just a means of calculating . The binomial series is one that recurs repeatedly and has become an extremely useful tool of analysis. It is a simple series that raises difficult questions. Foremost among these will be finding those values of x for which it converges.

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