Summation

S = 1 + 4r + 9r² + 16r³ + 25r⁴+ ..... (1)Sr = r + 4r² + 9r³ + 16r⁴+ 25r⁵ + ..... (2)
(1) - (2) :S - Sr = 1 + (4-1)r + (9-4)r² + (16-9)r³ + (25-16)r⁴+ ......S(1 - r) = 1 + 3r + 5r² + 7r³ + 9r⁴+ ...... (3)Sr(1 - r) = r + 3r² + 5r³ + 7r⁴+ ...... (4)
(3) - (4) :S(1 - r) - Sr(1 - r) = 1 + (3-1)r + (5-3)r² + (7-5)r³ + (9-7)r⁴+ ...... S(1 - r)² = 1 + 2(r + r² + r³ + r⁴+ ......)S(1 - r)² = 1 + 2( r / (1 - r) )S(1 - r)² = (1 + r) / (1 - r)S = (1 + r) / (1 - r)³