Question

How do you find the sum of the infinite geometric series 1 + 1/3 + 1/9 + ....?

Answer 1

Use the formula for the sum of an infinite geometric series to find:

`1+1/3+1/9+... = 3/2`

Explanation:

The general term of a geometric series is given by the formula:

`a_n = a r^(n-1)`

where `a` is the initial term and `r` the common ratio.

The sum of an infinite geometric series is given by the formula:

`sum_(n=1)^oo a r^(n-1) = a/(1-r)`

when `abs(r) < 1`. Otherwise the series does not converge.

For a derivation of this formula see: socratic.org/questions/how-do-you-find-the-sum-of-the-infinite-geometric-series-1-5-25-125

In our current example `a=1` and `r = 1/3`, so we find:

`sum_(n=1)^oo (1 * (1/3)^(n-1)) = 1/(1-1/3) = 1/(2/3) = 3/2`