If `S_(1), S_(2), S_(3),...,S_(n)` are the sums of infinite geometric series, whose first terms are 1, 2, 3,.., n and whose common rations are `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n+1)` respectively, then find the values of `S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)`.

Answer

`(1)/(6) (2n) (2n +1) (4n +1) -1`

Explanation

Consider an infinite GP with first term 1,2,3,.., n and common ratios `(1)/(2), (1)/(3), (1)/(4),..., (1)/(n +1)`
`:. S_(1) = (1)/(1 -1//2) = 2`
`{:(" "S_(2) = (2)/(1 -1//3) = 3),(" "vdots " "vdots " "vdots),(S_(2n -1) = (2n -1)/(1-1//2n) = 2n):}`
`:. S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ...+ S_(2n-1)^(2)`
`= 2^(2) + 3^(2) + 4^(2) + ...+ (2n)^(2)`
`= (1)/(6) (2n) (2n+1) (4n+1) -1`