Answer
You can find the infinite sum as follows:
${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$
Work Step by Step
The general form of the geometric sequence is$a,ar,a{{r}^{2}},a{{r}^{3}},\cdots $.
The formula of summation is given by:
${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{\left( 1-r \right)}$
Here $r\ne 0$ is the common ratio and $a$ is the scale factor or start value.
In an infinite geometric series, we can find the sum when $\left| r \right|<1$.
$\begin{align}
& {{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{\infty }} \right)}{\left( 1-r \right)} \\
& =\frac{{{a}_{1}}\left( 1-0 \right)}{\left( 1-r \right)} \\
& =\frac{{{a}_{1}}}{\left( 1-r \right)}
\end{align}$
Thus, the formula of summation for the infinite geometric series is given by
${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$
