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Evaluate the infinite geometric series:

\(2+\frac{1}{9}+\frac{1}{162}+\frac{1}{2916}+\dots\)

 

Thank You

 Dec 28, 2020
 #1
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Common ratio    2 * r = 1/9         r = 1/18

sum = a1 (1-r^n)/(1-r)

       = a1(1/(1-r) ) =   2 (1/(1-1/18)) = 2.12 

 Dec 28, 2020
 #2
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The common ratio between 2 terms in this geometric series is 1/18, and the first term of this geometric series is 2.

The sum of an infinite geometric series, if the common ratio is strictly greater than -1 and less than 1 and not equal to zero, is equal to \(a\over(1-r)\), where "a" is the first term and "r" is the common ratio.

Therefore, the answer to this is:

\(\frac{2}{1-\frac{1}{18}} = \frac{2}{\frac{17}{18}} = \frac{36}{17}\)

 Dec 28, 2020

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