mathematics-class-11-unit-10-chapter-06-sequence-and-series-l-6-10-vw8mdwkloy8

### <Success>Sum of n terms of a G.P.</Success>
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consider a G.P with first term " $a$ " and common ratio " $r$ "
$a, a r, a r^2, \ldots, a r^{n-1}, \ldots$
$ S_n=a+a r+a r^2+\cdots+a^{n-1}$
$\text { Let, }\quad r=1$
$S_n = \underbrace{a+a+ \cdots +a}_{n \ \text{times}}$
i.e $S_n=n a$.
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### <Success>Sequence of partial sum</Success>
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Given a sequence of real numbers $\{a_n\}_{n=1}^{\infty}$ the expression
$ a_1+a_2+\ldots = \sum_{n=1}^{\infty} a_n \ \ \text {is called a series. }$
Sequence of Partial sum
$S_n=a_1+a_2+\cdots+a_n$
$\left(S_n\right)_{n=1}^{\infty}$
$\operatorname{Lt}_{n \rightarrow \infty}\quad S_n=S$
$S=\sum_{n=1}^{\infty} a_n$
$\sum_{n=1}^{\infty} a_n \ $ is summable (convergent)
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### <Success>Convergence of the geometric series</Success>
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$ \underset{n \rightarrow \infty} {Lt} \quad S_n=\frac{a}{1-r} .$
$ \sum_{n=1}^{\infty} a r^{n-1}=\frac{a}{1-r}$
${\text { Let }|r|>1}$
$\underset{n \rightarrow \infty}{\operatorname{Lt}}{|r|^n}=\infty$
$\therefore \ \underset {{n \rightarrow \infty} } {\operatorname{Lt}} \quad S_n \text { does not exist }$
i.e. $\sum_{n=1}^{\infty} a r^{n-1}$ is NOT convergent
$(|r|>1)$
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