What is the sum of the series 2/3 − 2/9 + 2/27 − ....? (PAT Q1 2013)
This is the sum of a geometric series with an infinite number of terms.First, find the common ratio:(-2/9)/(2/3) = -1/3 , (2/27)/(-2/9) = -1/3Therefore, the common ratio is -1/3-1<(-1/3)<1 therefore the series will converge to a finite number. The general formula for a sum of an infinite geometric series is a/(1-r) where a is the first number in the sequence and r is the common ratio.So, substituting in the numbers, the sum of the series = (2/3)/(1-[-1/3]) = 1/2
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