### Taylor Expansions from India

FIg. 1: Brook Taylor (1685-1731). Image from NPG.

The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).

It is noteworthy that the series for ${\sin x}$, ${\cos x}$ and ${\arctan x}$ were known to mathematicians in India about 400 years before Taylor’s time.

Trigonometric Series

Several infinite series expansions were found during the seventeenth century. Generally, these were singular discoveries, without any broad theoretical basis. With Taylor’s work, they could all be seen to follow from his theorem.

We are familiar from elementary calculus with the Taylor series expansion of the trigonometric functions. For example, the sine and cosine expansions about ${x=0}$ are

$\displaystyle \begin{array}{rcl} \sin x &=& x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos x &=& 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \end{array}$

We can get a visual idea of how these series converge by considering the partial sums. We truncate the sine series to get a sequence of polynomials:

$\displaystyle \begin{array}{rcl} p_1(x) &=& \textstyle{x} \\ p_3(x) &=& \textstyle{x - \frac{x^3}{3!}} \\ p_5(x) &=& \textstyle{x - \frac{x^3}{3!} + \frac{x^5}{5!}} \\ p_7(x) &=& \textstyle{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}} \end{array}$

They approximate ${\sin x}$ better with increasing order. Fig. 2 shows the finite expansions ${p_1(x)}$, ${p_3(x)}$, ${p_5(x)}$ and ${p_7(x)}$. With four terms, a good approximation over a full wavelength is obtained.

Fig. 2: First four approximations over a full wavelength.

In the next figure, we show the expansions up to order 39, which approximates ${\sin x}$ over five wavelengths (only positive ${x}$ is shown in the figure).

FIg. 3: Higher truncations fit over greater ranges.

Taylor was by no means the first person to discover series expansions of transcendental functions. Gregory, Newton, Leibniz, Johann Bernoulli and de Moivre had all discovered special cases of his theorem. For example, in 1667 Gregory had derived the expansion

$\displaystyle \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$

This is the first infinite series expansion developed by a European mathematician.

The Gregory-Leibniz series for ${\pi}$

The arctan series with  ${x=1}$. gives the so-called Gregory-Leibniz series for ${\pi}$:

$\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \cdots$

However, the rate of convergence of this series is so slow that it is of no practical use in evaluating ${\pi}$. From the Table below we see that even with one million terms, only six digits of accuracy are obtained.

Convergence of Gregory-Leibniz series.

Early Discovery by Indian Mathematicians

It is remarkable that the series for ${\sin x}$, ${\cos x}$ and ${\arctan x}$ were known to the mathematicians of the Kerala School in Southern India, which flourished between AD 1300 and 1600. The arctangent series is also called the Madhava-Leibniz series, being a special case of a more general series expansion for the inverse tangent function, ascribed to the 14th century mathematician Madhava.

Madhava (c. 1340–1425), was a mathematician and astronomer from the town of Sangamagrama in Kerala, India. He is regarded as the founder of the Kerala school of astronomy and mathematics. He seems to have been the first person to use infinite series approximations to trigonometric functions.

Sources

${\bullet}$ Wikipedia article Brook Taylor.

${\bullet}$ Wikipedia article Madhava Series.

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