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Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

Taylor's Formula

>d'Alembert

Lagrange's Remainder

2.6 Emerging Doubts

2.5 Taylor Series (continued)

d'Alembert

One of the first mathematicans to study the convergence of series was Jean Le Rond d'Alembert in his paper of 1768, “Réflexions sur les suites et sur les racines imaginaires.” d'Alembert was science editor for Diderot's Encyclopédie and contributed many of the articles across many different fields. He was born Jean Le Rond, a foundling whose name was taken from the church of Saint Jean Le Rond in Paris on whose steps he had been abandoned. “Le Rond” (the round or plump) refers to the shape of the church. Perhaps feeling that John the Round was a name lacking in dignity, he added d'Alembert, and occasionally signed himself Jean Le Rond d'Alembert et de la Chapelle.

d'Alembert considered Newton's binomial series and asked when it is valid. In particular, he looked at the following series:

(2.5.2)

As d'Alembert pointed out, the series begins well. The partial sums of the first 100 and the first 101 terms are, respectively, 1.416223987 and 1.415756552. It appears to be converging very quickly toward the correct value near 1.41598098.

Starting out well is not enough. d'Alembert analyzed this series by comparing it to the geometric series. What characterizes a geometric series is the fact that the ratio of any two consecutive summands is always the same. This suggests analyzing the binomial series by looking at the ratio of consecutive summands. We can then compare our series to a geometric series. The series in (2.5.2) has as the nth summand

The absolute value of the ratio of consecutive summands is

d'Alembert now observed that this ratio is larger than 1 whenever n is larger than 300:

At n = 301, the ratio is larger than 1.000016, and it approaches 200/199 as n gets larger. Once we pass n=300, our summands will start to get larger. If the summands approach zero, we are not guaranteed convergence. On the other hand, if the summands do not approach zero, then the series cannot converge.

A similar analysis can be applied to the general binomial series,

The absolute value of the ratio of the n+1st summand to the nth summand is

As n increases, this ratio approaches |x|. If |x| > 1, then the summands do not approach 0 and the series cannot converge to . If |x| < 1, then the summands approach 0. Is this enough to guarantee that the binomial series converges to the desired value? d'Alembert did not answer this although he seemed to imply it. Neither did he investigate what happens when |x| = 1 (a question with a delicate answer that depends on the value of and the sign of x), or how far this approach can be extended to other series.

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