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QUESTION

What is the sum of the infinite geometric series 2+2/3+2/9+2/27+...?

##3##

a geometric series can be defined as: ##a_n = a_1*(r)^(n-1)## where ##a_1## is the first value of the series and ##r## is the common ratio.

The common ratio is ##1/3## and the first term is ##2## so our series is:

##a_n = 2*(1/3)^(n-1)##

You can only sum an infinite geometric series if it is <b>convergent</b>, that is, that it converges to one value. A series is convergent if ##abs(r) < 1## which in this case it is.

The equation to sum an is: ##a_1/(1-r)## so by plugging all values we get: ##2/(1-1/3)## which is ##2/(2/3)## or ##6/2## which simplifies to ##3##