سری هندسی

${\displaystyle S_n=\frac{a}{1-r}.}$

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

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

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{S}_n=\underset{n\to \mathrm{\infty }}{lim}(n+1)a=\pm \mathrm{\infty }}$

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

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

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

(٢) ${\displaystyle r{S}_n=ar+a{r}^2+...+a{r}^n+a{r}^{n+1}}$

(٣) ${\displaystyle S_n-r{S}_n=a-a{r}^{n+1}}$

${\displaystyle (1-r)S_n=a-a{r}^{n+1}}$

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

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{S}_n=\frac{a}{1-r}.}$