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I'm interested in understanding the history of the geometric series, especially how it was discovered and whether there exists continued fraction representations for the geometric series, just as there are continued fractions for many elementary functions in mathematics.

I would also appreciate some historical perspective on how geometric series played a role in the development of calculus.

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  • $\begingroup$ You can see the following link The Geometric Series Also you can visit mathcs.slu.edu/history-of-math/index.php/9.2_Geometric_series $\endgroup$ – Sachchidanand Prasad Sep 25 '15 at 9:12
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    $\begingroup$ Apparently, arithmetic/geometric series were already studied in Euclid, though not exactly under this terminology — “continually in proportion”. I don't know (and would like to know) when/where the terminology of arithmetic/geometric series has been invented. $\endgroup$ – ACL Nov 23 '18 at 9:10
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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. Nilkantha in his Aryabhatyabhasya.(Sastri 1970, commentary on Ganita 17, p. 142.)

एवं यासतुल्यच्छेदपरभागपरम्पराया अनन्ताया अपि संयोग

तस्यानंतानांपि कल्प्यामान्स्य योगस्याद्यावयविनः

परंपरांशच्छेदादेकोनच्छेदांशसामयं सर्वत्रापी समानमेव ।

which may be translated:(cf. Sarma 1972, p. 17) 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.

That is a+a + a +···= ad/d-1 .(It is assumed that the divisor d>1, so that the common ratio is less than 1.)

Reference: http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-1-final.pdf

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    $\begingroup$ Well... maybe infinite series do not originated in India, as Archimedes (III century BC) discussed in The Quadrature of the Parabola a geometric series with common ratio 1/4, while Nicole Oresme (1320-1382) discussed in Quaestiones super geometriam Euclidis convergence and divergence conditions for geometric series and obtained the sums of several geometric series (see e.g. here core.ac.uk/download/pdf/144470649.pdf) and also proved the divergence of the harmonic series. $\endgroup$ – user6530 Jan 5 at 13:33
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    $\begingroup$ To user6530's answer add that the general formula for the sum of a finite geometric series (not just the earlier specific examples from e.g. the Rhind Papyrus) is in Euclid's Elements Prop. IX.35. $\endgroup$ – David Eppstein Feb 5 at 1:22

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