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

Question
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.
0

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