Let `P_n=(2^3-1)/(2^3+1)*(3^3-1)/(3^3+1)*(4^3-1)/(4^3+1)....... (n^3-1)/(n^3+1)dot` Prove that `lim_(n->oo)P_n=2/3`

Let S_n=1+2+3++n and P_n=(S_2)/(S_2-1)*(S_3)/(S_3-1)*(S_4)/(S_4-1)* . . . *(S_n)/(S_n-1) Where n in N ,(ngeq2)dot Then lim_(n→oo)P_n=______

If 1/1^3 + (1+2)/(1^3+2^3)+(1+2+3)/(1^3+2^3+3^3) +.......n terms then lim_(n->oo) [S_n]

lim_(n->oo) (1.2+2.3+3.4+....+n(n+1))/n^3

If (1)/(1^(3))+(1+2)/(1^(3)+2^(3))+(1+2+3)/(1^(3)+2^(3)+3^(3))+......n terms then lim_(n rarr oo)[S_(n)]

the value of lim_(n->oo) {1/(n^3+1)+4/(n^3+1)+9/(n^3+1)+.................+n^2/(n^3+1)}

Let S_(n)=1+2+3+...+n and P_(n)=(S_(2))/(S_(2)-1)*(S_(3))/(S_(3)-1)*(S_(4))/(S_(4)-1)...(S_(n))/(S_(n)-1) Where n in N,(n>=2). Then lim_(x rarr oo)P_(n)=

if ^(2n+1)p_(n-1):^(2n-1)p_n=3:5 , prove that n = 4

lim_(n rarr oo){(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+....+(1)/((2n+1)(2n+3 ))

evaluate lim_ (n rarr oo) [(1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (2)) + ......... + (1) / (3 ^ (n))]