Question

Help!!! Geometric series.

Answer 1

Not just any geometric series, but an infinite one. The rare taming of infinity.

Since the ratio r=2/5 is strictly between -1 and 1 the infinite geometric series converges. The first term is also a=2/5 so the sum is

`S = (a)/(1-r) = (2/5)/(1 - 2/5) = \frac 2 3`

Answer: 2/3

That formula isn't too hard to derive. We have

`S = a + ar + ar^2 + ar^3 + ...`

`rS = ar + ar^2 + ar^3 + ...`

`S-rS = a`

`S = (a)/(1-r)`