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_(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_(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_(n) are the sums of infinite geometric series, whose first terms are 1,2,3,...n whose ratios are (1)/(2),(1)/(3),(1)/(4) ,...(1)/(n+1) respectively, then find the value of S_(1)^(2)+S_(2)^(2)+S_(3)^(2)+...+S_(2n-1)^(2) .

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_(n) are the sums to infinity of n infinte geometric series whose first terms are 1,2,3,… n and whose common ratios are (1)/(2), (1)/(3), (1)/(4), ….(1)/(n+1) respectively, show that, S_(1) + S_(2) + S_(3) +…S_(n) = (1)/(2)n(n+3)

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