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_(p) are the sums of infinite geometric series whose first terms are 1, 2, 3..... p and whose common ratios are (1)/(2),(1)/(3),.... (1)/(p+1) respectively, prove that S_(1) +S_(2)+S_(3)+.... + S_(p) = (1)/(2) p (p+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 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"}=

S_(1),S_(2), S_(3),...,S_(n) are sums of n infinite geometric progressions. The first terms of these progressions are 1, 2^(2)-1, 2^(3)-1, ..., 2^(n) – 1 and the common ratios are (1)/(2),(1)/(2^(2)),(1)/(2^(3)), ...., (1)/(2^(n)) . Calculate the value of S_(1), +S_(2),+ ... + S_(n).

If 1+ab+a^(2)b^(2)+a^(3)b^(3)+"..."+infty =(xy)/(x+y-1) are the sum of infinire geometric series whose first terms are 1,2,3,"….",p and whose common ratios are S_(1),S_(2),S_(3),"....,"S_(p) (1)/(2),(1)/(3),(1)/(4),"..."(1)/(p+1) respectively, prove that S_(1)+S_(2)+S_(3)+"....+"S_(p)=(p(p+3))/(2) .

If S_1,S_2, S_3, S_m are the sums of n terms of m A.P. ' s whose first terms are 1,2,3, ,m and common differences are 1,3,5, ,(2m-1) respectively. Show that S_1+S_2,+S_m=(m n)/2(m n+1)

If S_1,S_2, S_3, S_m are the sums of n terms of m A.P. ' s whose first terms are 1,2,3,....m and common differences are 1,3,5, ....(2m-1) respectively. Show that S_1+S_2.....+S_m=(m n)/2(m n+1)

Let S_(n),n=1,2,3,"…" be the sum of infinite geometric series, whose first term is n and the common ratio is (1)/(n+1) . Evaluate lim_(n to oo)(S_(1)S_(n)+S_(2)S_(n-1)+S_(3)S_(n-2)+"..."+S_(n)S_(1))/(S_(1)^(2)+S_(2)^(2)+"......"+S_(n)^(2)) .

Let S_(k) , where k = 1,2 ,....,100, denotes the sum of the infinite geometric series whose first term is (k -1)/(k!) and the common ratio is (1)/(k) . Then, the value of (100^(2))/(100!) +sum_(k=2)^(100) | (k^(2) - 3k +1) S_(k)| is....