Derivation of arithmetic-geometric series formula

S  = a + a^2 + a^3 + ...
aS =     a^2 + a^3 + ...
S - aS = a
S(1-a) = a
S = 1/(1-a) [the infinite geometric progression formula]

T  = a + 2 a^2 + 3 a^3 + ...
aT = a^2 + 2 a^3 + 3 a^4 + ...
aT + S = a + 2 a^2 + 3 a^3 + ...

T = aT + S
T(1-a) = S
T = S/(1-a) = 1/(1-a)^2