`lim_(x->oo) [(1)/(1times2)+(1)/(2times3)+(1)/(3times4)+.....+(1)/(n(n+1))]`

[(1)/(1times2)+(1)/(2times3)+(1)/(3times4)+......+(1)/(99times100)] =

lim_ (n rarr oo) [(1) / (1.2) + (1) / (2.3) + (1) / (3.4) + ... + (1) / (n (n + 1))] =

lim_(nto oo) {(1)/(n+1)+(1)/(n+2)+(1)/(n+3)+...+(1)/(n+n)} is, equal to

lim_ (n rarr oo) ((1) / (n + 1) + (1) / (n + 2) + (1) / (n + 3) + ...... + (1) / (6n ))

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

The value of lim_(x rarr oo)(2x^((1)/(2))+3x^((1)/(3))+4x^((1)/(4))+......+nx^((1)/(n)))/((2x-3)^((1)/(2))+(2x-3)^((1)/(3))+......+(2x-3)^((1)/(n))) is equal to

lim_ (x rarr oo) {(1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) + ...... (. 1) / (3 ^ (n))} =

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

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