# 1/4 + 1/16 + 1/64 + 1/256 + …

（重定向自1/4+1/16+1/64+1/256+…

${\displaystyle a+{\frac {a}{4}}+{\frac {a}{4^{2}}}+{\frac {a}{4^{3}}}+\cdots ={\frac {4}{3}}a.}$

## 圖像示範

3s = 1.

${\displaystyle 3\left({\frac {1}{4}}+{\frac {1}{4^{2}}}+{\frac {1}{4^{3}}}+{\frac {1}{4^{4}}}+\cdots \right)=1.}$
3s = 1

## 阿基米德

${\displaystyle A+B+C+D+\cdots +Z+{\frac {1}{3}}Z={\frac {4}{3}}A.}$

${\displaystyle {\begin{array}{rcl}\displaystyle B+C+\cdots +Z+{\frac {B}{3}}+{\frac {C}{3}}+\cdots +{\frac {Z}{3}}&=&\displaystyle {\frac {4B}{3}}+{\frac {4C}{3}}+\cdots +{\frac {4Z}{3}}\\[1em]&=&\displaystyle {\frac {1}{3}}(A+B+\cdots +Y).\end{array}}}$

${\displaystyle {\frac {B}{3}}+{\frac {C}{3}}+\cdots +{\frac {Y}{3}}={\frac {1}{3}}(B+C+\cdots +Y).}$

${\displaystyle B+C+\cdots +Z+{\frac {Z}{3}}={\frac {1}{3}}A}$

${\displaystyle 1+{\frac {1}{4}}+{\frac {1}{4^{2}}}+\cdots +{\frac {1}{4^{n}}}={\frac {1-\left({\frac {1}{4}}\right)^{n+1}}{1-{\frac {1}{4}}}}.}$

## 參考

1. ^ Shawyer and Watson p. 3.
2. Nelsen and Alsina p. 74.
3. ^ Ajose and Nelson.
4. ^ Stein p. 46.
5. ^ Mabry.
6. ^ This presentation is a shortened version of Heath p.250.

## 參考書目

• Ajose, Sunday and Roger Nelsen. Proof without Words: Geometric Series. Mathematics Magazine. June 1994, 67 (3): 230. JSTOR 2690617. doi:10.2307/2690617.
• Heath, T. L. The Works of Archimedes. Cambridge UP. 1953 [1897]. Page images at Casselman, Bill. Archimedes' quadrature of the parabola. [2007-03-22]. HTML with figures and commentary at Otero, Daniel E. Archimedes of Syracuse. 2002 [2007-03-22]. （原始内容存档于7 March 2007）.
• Mabry, Rick. Proof without Words: 14 + (14)2 + (14)3 + ⋯ = 13. Mathematics Magazine. February 1999, 72 (1): 63. JSTOR 2691318.
• Nelsen, Roger B.; Alsina, Claudi. Math Made Visual: Creating Images for Understanding Mathematics. MAA. 2006. ISBN 0-88385-746-4.
• Shawyer, Bruce; Watson, Bruce. Borel's Methods of Summability: Theory and Applications. Oxford UP. 1994. ISBN 0-19-853585-6.
• Stein, Sherman K. Archimedes: What Did He Do Besides Cry Eureka?. MAA. 1999. ISBN 0-88385-718-9.
• Swain, Gordon; Dence, Thomas. Archimedes' Quadrature of the Parabola Revisited. Mathematics Magazine. April 1998, 71 (2): 123–30. JSTOR 2691014. doi:10.2307/2691014.