Find `lim_(n->oo) [n/((n + 1) sqrt(2n +1)) +n/((n +2) sqrt(2(2n +2))) +n/((n +3) sqrt(3(2n +3) ) +...] ` upto n terms

lim_(n->oo) ((sqrt(n^2+n)-1)/n)^(2sqrt(n^2+n)-1)

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

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

lim_ (n rarr oo) ((sqrt (n ^ (2) + n) -1) / (n)) ^ (2sqrt (n ^ (2) + n) -1)

lim_ (n rarr oo) (3) / (n) [1 + sqrt ((n) / (n + 3)) + sqrt ((n) / (n + 6)) + sqrt ((n) / (n +9)) + ...... + sqrt ((n) / (n + 3 (n-1)))

Find lim_ (n rarr oo) [(n ^ (3) +1) ^ ((1) / (2)) - n ^ ((3) / (2))] -: n ^ ((3) / ( 2))

lim_(n to oo) (sqrt(1) + sqrt(2) + …… + sqrt(n))/(n^(3//2)) equals :

lim_(n->oo)[1/sqrt(2n-1^2) +1/sqrt(4n-2^2)+1/sqrt(6n-3^2)+...+1/n]

Find underset( n rarr oo) ("lim") (sqrt( n^(2) + 1)+ sqrt( n ) )/( sqrt( n ^(2) + 1)- sqrt( n ) )