Find the sum of the infinite geometric series. sum_{i=1}^infty12(-0.7)^{i-1}

To find:
The sum of the infinite series $\sum _{i=1}^{\mathrm{\infty }}12(-0.7{)}^{i-1}$
Concept used:
The sum of infinite geometric series can be obtained by the formula,
$S_{\mathrm{\infty }}=\frac{a}{1-r}(r<1)$
Here, a is first term, r is common ratio and $S_{\mathrm{\infty }}$ is sum of infinite geometric series.
Calculation:
Expand the summation as follows:
$\sum _{i=1}^{\mathrm{\infty }}12(-0.7{)}^{i-1}=12(-0.7{)}^{1-1}+12(-0.7{)}^{2-1}+12(-0.7{)}^{3-1}+...$
$=12+12(-0.7)+12(-0.7{)}^2+...$
It is observed from the series that the first term is 12 and common ratio is (−0.7) and common ratio is less than 1.
Substitute 12 for a and (−0.7) for r in equation (1).
$S_{\mathrm{\infty }}=\frac{12}{10(-0.7)}$
$=\frac{12}{1+0.7}$
$=\frac{12}{1.7}$
$=7.06$
Thus, the sum of the infinite series is 7.06.