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)`.

If S_(1), S_(2),cdots S_(n) , ., are the sums of infinite geometric series whose first terms are 1, 2, 3,…… ,n and common ratios are (1)/(2) ,(1)/(3),(1)/(4),cdots,(1)/(n+1) then S_(1)+S_(2)+S_(3)+cdots+S_(n) =

If S_1, S_2, S_3,…..S_p are the sum of infinite geometric series whose first terms are 1,2,3,…p and whose common ratios are 1/2,1/3,1/4,….1/(p+1) respectively, prove that S_1+S_2+……..+S_p=p(p+3)/2

If S_(1),S_(2),S_(3),"…….",S_(2n) are the same of infinite geometric series whose first terms are respectively 1 , 2, 3,"…….", 2n and common ratio are respectively. 1/2, 1/3, '…..". (1)/(2n+1) .Find the value of S_(1)^(2) + S_(2)^(2) + "……" + S_(2n-1)^(2) .

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 ratio 1/2,1/3,1/4,........,1/(n+1),....respectively, then find the value of sum_(r=1)^(2n-1) S_1^2.

If S_(1), S_(2), S_(3),….., S_(n) are the sum of infinite geometric series whose first terms are 1,3,5…., (2n-1) and whose common rations are 2/3, 2/5,…., (2)/(2n +1) respectively, then {(1)/(S_(1) S_(2)S_(3))+ (1)/(S_(2) S_(3) S_(4))+ (1)/(S_(3) S_(4)S_(5))+ ........."upon infinite terms"}=

If S_1, S_2,S_3,S_4,.....,S_p denotes the sums of infinite geometric series whose first terms are 1, 2, 3,...p respectively and whose common ratios are 1/2,1/3,1/4,.....1/(p+1) respectively .then S_1+S_2+S_3+....+S_p = kp(p+3), where k =?