Geometric series

In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series ${\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+\cdots$ is a geometric series with common ratio Template:Tmath, which converges to the sum of Template:Tmath. Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors.

Quotes about Geometric series

• The first known mention of the geometric series is due to Mahavira (850 CE) in his Ganitasarasangraha (ii.93): पदमितगुणहतिगुणितप्रभवस््स्ययाद्णगु धनं तदाद्यूनम।् एकोनगुणविभक्तं गुणसङ्कलितं विजानीयात्॥ which translates to
  The first term when multiplied by the continued product of the common ratio, taken as many times as the number of terms, gives rise to the gunadhana [the (n+1)th term of the geometric progression]. And it has to be understood that this gunadhana, when diminished by the first term and then divided by the common ratio lessened by one, becomes the sum of the series in the geometric progression.
  • Mahāvīra, quoted in Singh, A. N. ‘On the Use of Series in Hindu Mathematics’. Osiris Vol. 1 (Jan., 1936), pp. 606-628.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• The sum of an infinite [anantya] series, whose later terms (after the first) are got by dividing the preceding one by the same divisor everywhere, is equal to the first term multiplied by the common divisor, and divided by one less than the common divisor.
  • 15th-16th c. Nı̄lakant.ha in his Āryabhatı̄yabhāsya.
  • The infinite series had originated in India by the 14th c. An explicit formula for the sum of an infinite (anantya) geometric series is given by the 15th-16th c. Nı̄lakantha in his Āryabhatı̄yabhāsya.(Sastri 1970, commentary on Ganita 17, p. 142.) which may be translated:(cf. Sarma 1972, p. 17) Sarma, K. V., 1972, A History of the Kerala School of Astronomy (in perspective), Panjab Univeristy, Hoshiarpur.
  • quoted from Raju, C. K. ‘Calculus’. In Encyclopedia of Non-Western Science, Technology and Medicine, edited by Helaine Selin, 1010–15. Dordrecht: Springer, 2016.