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Subtracting s n from both sides,. As n approaches infinity, s n tends to 1. The tortoise, with a 10 meter advantage, Zeno argued, would win. The Achilles would have to move 10 meters to catch up to the tortoise, but by then, the tortoise would already have moved another five meters.
Achilles would then have to move 5 meters, where the tortoise would move 2. The parts of the Eye of Horus were once thought to represent the first six summands of the series.
From Wikipedia, the free encyclopedia. Archived from the original on Cauchy sequence Monotone sequence Periodic sequence.
Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping series.
Archimedes encounters the series in his work Quadrature of the Parabola. To get there, he takes a break from parabolas to introduce an algebraic lemma:.
Given a series of areas A , B , C , D , … , Z , of which A is the greatest, and each is equal to four times the next in order, then .
The same strategy works for any finite geometric series. He does not quite  take the limit of the above partial sums, but in modern calculus this step is easy enough:.
From Wikipedia, the free encyclopedia. For example, Shawyer and Watson p. Ajose, Sunday and Roger Nelsen June The Works of Archimedes.
Page images at Casselman, Bill. Archived from the original on 7 March Mabry, Rick February