Explanation:
sum of (2^n / 5^n), with n from 1 to infinity.
an = (2^n / 5^n)
an+1 = (2^(n+1) / 5^(n+1)) = an × 2/5
so, we can use the Ratio Test (d'Alembert's criterion) to prove that this series is convergent :
r = lim | (an+1 / an) |, n going to infinity.
if r < 1, then the series is convergent.
an+1/an = (2^(n+1) / 5^(n+1)) / (2^n / 5^n) = 2/5.
therefore
r = lim | (an+1 / an) | = 2/5, n going to infinity.
2/5 < 1, therefore the series is convergent.
the sum of an infinite geometric series (like this one) is
a1/(1 - r)
our a1 = 2/5.
and NOT 2⁰/5⁰ = 1 !
and r is also 2/5.
so we have
sum of the series = 2/5 / (1 - 2/5) = 2/5 / 3/5 =
= (2×5) / (5×3) = 2/3