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 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,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 =?

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 sums of infinite geometric series whose first terms are respectively 1,2,3,…2n and common ratioi 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_q are the sums of n terms of q ,AP's whose first terms are 1,2,3,...........q and common difference are 1,3,5,...........,(2q-1) respectively ,show that S_1 + S_2 + S_3..........+S_q=1/2nq(nq+1)

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_(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"}=