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

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Answer 1 · 2015-11-27T07:37:23

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

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

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