Precalculus (6th Edition) Blitzer

You can find the infinite sum as follows: ${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$
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 ${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$