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_(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_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 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 1+ab+a^(2)b^(2)+a^(3)b^(3)+"..."+infty =(xy)/(x+y-1) are the sum of infinite 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 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 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),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) =