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
[tex]S = \dfrac{a}{1-r} = \dfrac{2/5}{1 - 2/5} = \dfrac 2 3[/tex]
Answer: 2/3
That formula isn't too hard to derive. We have
[tex]S = a + ar + ar^2 + ar^3 + ...[/tex]
[tex]rS = ar + ar^2 + ar^3 + ...[/tex]
[tex]S-rS = a[/tex]
[tex]S = \dfrac{a}{1-r}[/tex]
