سری هندسی

${\displaystyle S_{n}\;=\;{\frac {a}{1-r}}.}$

${\displaystyle a+a+a+a+....}$

${\displaystyle S_{n}=(n+1)a}$

${\displaystyle \lim _{n\to \infty }S_{n}=\lim _{n\to \infty }(n+1)a=\pm \infty }$

${\displaystyle a-a+a-a+....}$

${\displaystyle a,0,a,0,a,...}$

(١) ${\displaystyle S_{n}=a+ar+ar^{2}+...+ar^{n}}$

(٢) ${\displaystyle rS_{n}=ar+ar^{2}+...+ar^{n}+ar^{n+1}}$

(٣) ${\displaystyle S_{n}-rS_{n}=a-ar^{n+1}}$

${\displaystyle (1-r)S_{n}\;=\;a-ar^{n+1}}$

${\displaystyle S_{n}\;=\;{\frac {a-ar^{n+1}}{1-r}}={\frac {a}{1-r}}(1-r^{n+1})}$

${\displaystyle \lim _{n\to \infty }S_{n}\;=\;{\frac {a}{1-r}}.}$