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

$S = \dfrac{a}{1-r} = \dfrac{2/5}{1 - 2/5} = \dfrac 2 3$

Answer: 2/3

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

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

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

$S-rS = a$

$S = \dfrac{a}{1-r}$