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

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Series
asked 2021-02-09
Find the sum of the infinite geometric series.
$$\sum_{i=1}^\infty12(-0.7)^{i-1}$$

## Answers (1)

2021-02-10
To find:
The sum of the infinite series $$\sum_{i=1}^\infty12(-0.7)^{i-1}$$
Concept used:
The sum of infinite geometric series can be obtained by the formula,
$$S_\infty=\frac{a}{1-r}\quad(r<1)$$</span>
Here, a is first term, r is common ratio and $$S_\infty$$ is sum of infinite geometric series.
Calculation:
Expand the summation as follows:
$$\sum_{i=1}^\infty12(-0.7)^{i-1}=12(-0.7)^{1-1}+12(-0.7)^{2-1}+12(-0.7)^{3-1}+...$$
$$=12+12(-0.7)+12(-0.7)^2+...$$
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.

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