Question

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

Answer 1

`3`

Explanation:

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 infinite series 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`