Let `S_n,n=1,2,2....` be the sum of infinite geometric series whose first term is n and the common ratio is . `1/(n+1).`Evaluate `Lim_(n->oo) (S_1S_n+S_2S_(n-1)+S_3S_(n-2)+.....+S_nS_1)/(S_1^2+S_2^2+....+S_n^2)`

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_(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_(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_(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)) .

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_(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_(n) denotes the sum of n terms of a G.P whose first term is a and common ratio is r then find the sum of S_(1),S_(3),S_(5).........,S_(2)n-1

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).