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. ^ 2.0 2.1 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.