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

Find the sum of the infinite geometric series.
$\sum _{i=1}^{\mathrm{\infty }}12\left(-0.7{\right)}^{i-1}$
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To find:
The sum of the infinite series $\sum _{i=1}^{\mathrm{\infty }}12\left(-0.7{\right)}^{i-1}$
Concept used:
The sum of infinite geometric series can be obtained by the formula,
${S}_{\mathrm{\infty }}=\frac{a}{1-r}\phantom{\rule{1em}{0ex}}\left(r<1\right)$
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:
$\sum _{i=1}^{\mathrm{\infty }}12\left(-0.7{\right)}^{i-1}=12\left(-0.7{\right)}^{1-1}+12\left(-0.7{\right)}^{2-1}+12\left(-0.7{\right)}^{3-1}+...$
$=12+12\left(-0.7\right)+12\left(-0.7{\right)}^{2}+...$
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).
${S}_{\mathrm{\infty }}=\frac{12}{10\left(-0.7\right)}$
$=\frac{12}{1+0.7}$
$=\frac{12}{1.7}$
$=7.06$
Thus, the sum of the infinite series is 7.06.

Jeffrey Jordon