The Infinite series of Pi and it’s approximation by Madhava

The infinite series for pi is mostly today known as Leibniz formula for π. But many few people know that this series was already discovered in India by Madhava (c. 1340–1425 AD) of Sangamagrama, 300 years before Leibniz or Gregory. Although none of the Madhava’s works have survived but most of the series attributed to him can be found in the books of his students and at many places these authors have clearly stated that these series are “As told by Madhava”.

The unique thing about the series given by Madhava is that he gave this series in form of a beautiful verse.

Infinite series for pi — as given in Kriyakramakari :

This translates to

“The diameter multiplied by four and divided by unity (is found and stored). Again, the products of the diameter and four are divided by the odd numbers like 3,5 etc., and the results are subtracted and added in order (to earlier stored result).”

This in other words is:

This was a brilliant contribution by Madhava, in fact Leibniz was highly praised by the western world for coming up with this infinite series.

Madhava’s correction for pi

What’s wonderful about Madhava was that he even provided the correction term for the infinite series pi, obtaining fast convergent approximations for the series.

This result is cited in Yuktidipikä of Sankara Variyar.

This translates to:

“The diameter multiplied by four and divided by unity (is found and stored). Again, the products of the diameter and four are divided by the odd numbers like 3,5 etc., and the results are subtracted and added in order (to earlier stored result)

Take half of the succeeding even number as the multiplier at whichever (odd) number the division process is stopped, because of boredom (by the slow converging process). The square of that (even number) added to unity is the divisor. The ratio has to be multiplied by the product of the diameter and four as (stated) earlier.

The result obtained has to be added if the earlier term (in the series) has been subtracted and subtracted if the earlier term has been added. The resulting circumference is very accurate; in fact, more accurate than the one which may be obtained by continuing the division process (with a large number of terms in the series).”

In other words, it is

Describing different ways by which better approximations can be obtained,
finally Sankara Variyar states a more accurate correction term:

“Now I shall write of certain other correction more accurate than this. In the last term the multiplier should be the square of half the even number together with one, and the divisor four times that, added by unity, and then multiplied by half the even number. After division by the odd numbers 3, 5 etc, the final operation must be made as just indicated”

This shows that Madhava quite well knew this series 300 years before Leibniz and Gregory. This is also evident from his attempt to devise a value of π with the help of a slowly converging series and ways to improve the result by suggesting corrections after a limited number of steps to attain the result much faster.

If today a series is to be given a name that actually honors the discoverer then it should be named Madhava series for pi and not Leibniz- Madhava series or anything else.

References:

Indian Journal of History of Science, 47.4 (2012)

Algorithms in Indian Mathematics, M.S.Sriram

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