1/4 + 1/16 + 1/64 + 1/256 + · · ·

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 and Watson p. 3.] Its sum is 1/3. More generally, for any "a", the infinite geometric series whose first term is "a" and whose common ratio is 1/4 is convergent with the sum:$a+frac\{a\}\{4\}+frac\{a\}\{4^2\}+frac\{a\}\{4^3\}+cdots\; =\; frac43\; a.$

Visual demonstrations

The series 1/4 + 1/16 + 1/64 + 1/256 + · · · lends itself to some particularly simple visual demonstrations 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 and Alsina p. 74.] Ajose and Nelson.] 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:$3left(frac14+frac\{1\}\{4^2\}+frac\{1\}\{4^3\}+frac\{1\}\{4^4\}+cdots\; ight)\; =\; 1.$

Archimedes' own illustration, adapted at top, [Heath p.250] was slightly different, being closer to the equation

:$frac34+frac\{3\}\{4^2\}+frac\{3\}\{4^3\}+frac\{3\}\{4^4\}+cdots\; =\; 1.$See below for details on Archimedes' interpretation.

The same geometric strategy also works for triangles, as in the figure on the right:Stein p. 46.] Mabry.] 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 Sierpinski triangle. [Nelson and Alsina p.56]

Archimedes

Archimedes encounters the series in his work "Quadrature of the Parabola". He is finding 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 in that the total area is 4/3 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, then [This is a quotation from Heath's English translation (p.249).] :$A\; +\; B\; +\; C\; +\; D\; +\; cdots\; +\; Z\; +\; frac13\; Z\; =\; frac43\; A.$

Archimedes proves the proposition by first calculating:On the other hand,:$frac\{B\}\{3\}+frac\{C\}\{3\}+cdots+frac\{Y\}\{3\}\; =\; frac13(B+C+cdots+Y).$

Subtracting this equation from the previous equation yields:$B+C+cdots+Z+frac\{Z\}\{3\}\; =\; frac13\; A$and adding "A" to both sides gives the desired result. [This presentation is a shortened version of Heath p.250.]

Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series nowrap|1 + 1/4 + 1/16 + · · · are::$1+frac\{1\}\{4\}+frac\{1\}\{4^2\}+cdots+frac\{1\}\{4^n\}=frac\{1-left(frac14\; ight)^\{n+1\{1-frac14\}.$

This form can be proved by multiplying both sides by 1 − 1/4 and observing that most 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 "quite" [Modern authors differ on how appropriate it is to say that Archimedes summed the infinite series. For example, Shawyer and Watson (p.3) simply say he did; Swain and Dence say that "Archimedes applied an indirect limiting process"; and Stein (p.45) stops short with the finite sums.] take the limit of the above partial sums, but in modern calculus this step is easy enough::$lim\_\{n\; oinfty\}\; frac\{1-left(frac14\; ight)^\{n+1\{1-frac14\}\; =\; frac\{1\}\{1-frac14\}\; =\; frac43.$

Since the sum of an infinite series is defined as the limit of its partial sums,:$1+frac14+frac\{1\}\{4^2\}+frac\{1\}\{4^3\}+cdots\; =\; frac43.$

Notes

References

*cite journal |title=Proof without Words: Geometric Series |author=Ajose, Sunday and Roger Nelsen |journal=Mathematics Magazine |volume=67 |issue=3 |month=June |year=1994 |pages=230 |url=http://links.jstor.org/sici?sici=0025-570X%28199406%2967%3A3%3C230%3APWWGS%3E2.0.CO%3B2-A
*cite book |first=T. L. |last=Heath |title=The Works of Archimedes |year=1953 |origyear=1897 |publisher=Cambridge UP Page images at cite web |first=Bill |last=Casselman |title=Archimedes' quadrature of the parabola |url=http://www.math.ubc.ca/~cass/archimedes/parabola.html |accessdate=2007-03-22 HTML with figures and commentary at cite web |first=Daniel E. |last=Otero |year=2002 |title=Archimedes of Syracuse |url=http://www.cs.xu.edu/math/math147/02f/archimedes/archpartext.html |accessdate=2007-03-22
*cite journal |title=Proof without Words: ¹⁄₄ + (¹⁄₄)² + (¹⁄₄)³ + · · · = ¹⁄₃ |first=Rick |last=Mabry |journal=Mathematics Magazine |volume=72 |issue=1 |month=February |year=1999 |pages=63 |url=http://links.jstor.org/sici?sici=0025-570X%28199902%2972%3A1%3C63%3APWW%3C%3E2.0.CO%3B2-O
*cite book |author=Nelsen, Roger B. and Claudi Alsina |title=Math Made Visual: Creating Images for Understanding Mathematics |year=2006 |publisher=MAA |id=ISBN 0883857464
*cite book |author=Shawyer, Bruce and Bruce Watson |title=Borel's Methods of Summability: Theory and Applications |publisher=Oxford UP |year=1994 |id=ISBN 0-19-853585-6
*cite book |first=Sherman K. |last=Stein |title=Archimedes: What Did He Do Besides Cry Eureka? |publisher=MAA |year=1999 |id=ISBN 0883857189
*cite journal |author=Swain, Gordon and Thomas Dence |title=Archimedes' Quadrature of the Parabola Revisited |journal=Mathematics Magazine |volume=71 |issue=2 |month=April |year=1998 |pages=123–30 |url=http://links.jstor.org/sici?sici=0025-570X%28199804%2971%3A2%3C123%3AAQOTPR%3E2.0.CO%3B2-Q