1/4 + 1/16 + 1/64 + 1/256 + ⋯

Infinite series summable to 1/3 right|thumb|Archimedes' figure with a = 3 4 In mathematics, the infinite series 1 4 + 1 16 + 1 64 + 1 256 + ⋯ is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.Shawyer Watson 1994 3 As it is a geometric series with first term 1 4 and common ratio 1 4, its sum is Visual demonstrations left|thumb|3s = 1. The series 1 4 + 1 16 + 1 64 + 1 256 + ⋯ lends itself to some particularly simple visual proofs because a square and a triangle both divide into four similar pieces, each of which contains 1 4 the area of the original. In the figure on the left, Nelsen Alsina 2006 74 Ajose Nelsen 1994 230 if the large square is taken to have area 1, then the largest black square has area 1 2 × 1 2 = 1 4. Likewise, the second largest black square has area 1 16, and the third largest black square has area 1 64. The area taken up by all of the black squares together is therefore 1 4 + 1 16 + 1 64 + ⋯, and this is also the area taken up by the gray squares and the white squares. Since these three areas cover the unit square, the figure demonstrates that Archimedes' own illustration, adapted at top,Heath 1953 250 was slightly different, being closer to the equation right|thumb|3s = 1 again See below for details on Archimedes' interpretation. The same geometric strategy also works for triangles, as in the figure on the right: Nelsen Alsina 2006 74 Stein 1999 46 Mabry 1999 63 if the large triangle has area 1, then the largest black triangle has area 1 4, and so on. The figure as a whole has a self-similarity between the large triangle and its upper sub-triangle. A related construction making the figure similar to all three of its corner pieces produces the Sierpiński triangle.Nelsen Alsina 2006 56 Proof by Archimedes 250px|thumb|This curve is a parabola. The dots on the secant line AE are equally spaced. Archimedes showed that the sum of the areas of triangles ABC and CDE is 1 4 of the area of triangle ACE. He then constructs another layer of four triangles atop those, the sum of whose areas is 1 4 of the sum of the areas of ABC and CDE, and then another layer of eight triangles atop that, having 1 4 of that area, and so on. He concluded that the area between the secant line and the curve is 4 3 the area of triangle ACE. Archimedes encounters the series in his work Quadrature of the Parabola. He finds the area inside a parabola by the method of exhaustion, and he gets a series of triangles; each stage of the construction adds an area 1 4 times the area of the previous stage. His desired result is that the total area is 4 3 times the area of the first stage. To get there, he takes a break from parabolas to introduce an algebraic lemma: Proposition 23. 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, thenThis is a quotation from the English translation of Heath 1953 249. Archimedes proves the proposition by first calculating On the other hand, Subtracting this equation from the previous equation yields and adding A to both sides gives the desired result.This presentation is a shortened version of Heath 1953 250. Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1 4 + 1 16 + ⋯ are: This form can be proved by multiplying both sides by 1 − 1 4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs. The same strategy works for any finite geometric series. The limit Archimedes' Proposition 24 applies the finite (but indeterminate) sum in Proposition 23 to the area inside a parabola by a double reductio ad absurdum. He does not quiteModern authors differ on how appropriate it is to say that Archimedes summed the infinite series. For example, Shawyer Watson 1994 3 simply say he did; Swain and Dence say that "Archimedes applied an indirect limiting process"; and Stein 1999 45 stops short with the finite sums. take the limit of the above partial sums, but in modern calculus this step is easy enough: Since the sum of an infinite series is defined as the limit of its partial sums, Notes References 2 Ajose Sunday Nelsen Roger Proof without Words: Geometric Series 10.2307/2690617 Mathematics Magazine 67 3 230 2690617 June 1994 Heath T. L. The Works of Archimedes 1953 1897 Cambridge University Press Page images at Casselman Bill Archimedes' quadrature of the parabola http://www.math.ubc.ca/~cass/archimedes/parabola.html 2007-03-22 2012-03-20 https://web.archive.org/web/20120320124529/http://www.math.ubc.ca/~cass/archimedes/parabola.html dead HTML with figures and commentary at Otero Daniel E. 2002 Archimedes of Syracuse http://www.cs.xu.edu/math/math147/02f/archimedes/archpartext.html 2007-03-22 https://web.archive.org/web/20070307141104/http://www.cs.xu.edu/math/math147/02f/archimedes/archpartext.html 7 March 2007 Mabry Rick Proof without Words: . Mathematics Magazine 72 1 63 2691318 February 1999 10.1080/0025570X.1999.11996702 Nelsen Roger B. Alsina Claudi Math Made Visual: Creating Images for Understanding Mathematics 2006 Mathematics Association of America 0-88385-746-4 Shawyer Bruce Bruce Watson Borel's Methods of Summability: Theory and Applications Oxford University Press 1994 0-19-853585-6 Stein Sherman K. Sherman K. Stein Archimedes: What Did He Do Besides Cry Eureka? Mathematics Association of America 1999 0-88385-718-9 registration https://archive.org/details/archimedeswhatdi00stei Swain Gordon Dence Thomas Archimedes' Quadrature of the Parabola Revisited Mathematics Magazine 71 2 123–30 10.2307/2691014 2691014 April 1998 Category:Geometric series Category:Proof without words Category:4 (number)