If $\dfrac{t_{n+1}}{t_n}=r$ (constant), for $n=1,2,3,...$ then $(t_n)$ is called a Geometric Sequence or Geometric Progression (G.P). and the series $\sum_{n=1}^{\infty }t_n$ is called a geometric series. The constant $r$ is known as the common ratio $(c.r)$ .

Note -:

(i) If $t_1\text{( first term )}=a$ , common ratio $=r$ , then $t_n = ar^{n-1}$

(ii) No term of a GP can be zero, for otherwise $\dfrac{t_{n+1}}{t_n}$ will be meaningless for the corresponding value of $n$ .

$n^{th}$ Partial Sum of a Geometric Series -:

For a geometric series with $t_n=a$ and common ratio ratio $=r$ , $S_n=t_1+t_2+...+t_{n-1}=a+ar+ar^2+...+ar^{n-1}$

$\therefore rS_n=ar+ar^2+...+ar^{n-1}+ar^n$

$\implies \text{(Subtracting)}(1-r)S_n = a (1-r^n)$

$\therefore S_n=\dfrac{a(1-r^n)}{1-r}$ , for $r\neq 1$

If $r=1$ , then $t_n=a$ , for every $n$ , so that $S_n=na$

Sum of Geometric Series -:

If $|r|$ < $1$ , i.e. $-1$ < $r$ < $1$ then $r^n\to 0$ . When $n\to \infty$ . So for the geometric series with $|r|$ < $1$ .

We have $\lim_{n \to \infty } S_n=\lim_{n \to \infty } \ a \cdot\dfrac{1-r^n}{1-r}=\dfrac{a}{1-r}$

Therefore $\sum_{n=1}^{\infty }ar^{n-1}=\dfrac{a}{1-r}$ ; if $\left| r \right|$ < $1$ and $\sum_{n=1}^{\infty }ar^{n-1}$ diverges if $\left| r \right|> 1$ .

Hence the geometric series $\sum_{n=1}^{\infty }ar^{n-1}$ converges if $\left| r \right|$ < $1$ and diverges if $\left| r \right| >1$ .

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