Question

?How do you find the sum of the infinite geometric series `2^n/5^(2n+1)`?

Answer 1

`sum_(n=0)^(oo)2^n/5^(2n+1) = 5/23`

Explanation:

The sum of an infinite geometric series with ratio `r` where `|r| < 1` is given by
`sum_(n=0)^(oo)ar^n = a/(1-r)`

(A quick derivation for this formula is included in this answer: Can an infinite series have a sum? )

Now
`2^n/5^(2n+1) =(1/5)(2^n/5^(2n)) = (1/5)(2^n/25^n) = (1/5)(2/25)^n`

So, as `|2/25| < 1`

we have
`sum_(n=0)^(oo)2^n/5^(2n+1) = sum_(n=0)^(oo)(1/5)(2/25)^n = (1/5)/(1-(2/25)) = 5/23`