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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