Open Question: What is the exact sum of this series?

(2^n + 3^n) / 5^n =>
(2^n / 5^n) + (3^n / 5^n) =>
(2/5)^n + (3/5)^n

sum( (2/5)^n + (3/5)^n , n = 0 , n = inf) =>
sum((2/5)^n , n = 0 , n = inf) + sum((3/5)^n , n = 0 , n = inf) =>
S + T

S = (2/5)^0 + (2/5)^1 + .... + (2/5)^m
S * (2/5) = (2/5) + (2/5)^2 + ... + (2/5)^m + (2/5)^(m + 1)
S - S * (2/5) = (2/5)^0 - (2/5) * (2/5)^m
m => inf
S * (1 - 2/5) = (2/5)^0 - (2/5) * (2/5)^inf
S * (3/5) = 1 - (2/5) * 0
S * (3/5) = 1
S = 5/3

T = (3/5)^0 + (3/5)^1 + ... + (3/5)^m
T * (3/5) = (3/5) + (3/5)^2 + (3/5)^m + (3/5) * (3/5)^m
m => inf
T - T * (3/5) = (3/5)^0 - (3/5) * (3/5)^inf
T * (1 - (3/5)) = 1 - (3/5) * 0
T * (2/5) = 1
T = 5/2

S + T =>
(5/3) + (5/2) =>
(10/6) + (15/6) =>
(25/6)

25/6 is the sum

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