Find the sum of the infinite geometric series. ∑i=1∞12(−0.7)i−1

Question

Find the sum of the infinite geometric series.  ∑i=1∞12(−0.7)i−1

toroztatG · Accepted Answer

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

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

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

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