`lim_(n->oo)(1^3+2^3++n^3)/((n-1)^4)`

Evaluate : lim_(n-> oo) (1^4+2^4+3^4+...+n^4)/n^5 - lim_(n->oo) (1^3+2^3+...+n^3)/n^5

Evaluate the following limit: (lim)_(n rarr oo)(1^(3)+2^(3)+n^(3))/((n-1)^(4))

lim_(n to oo) [ 1^2/n^3 + (2^2)/(n^3) + …+ ((n-1)^2)/(n^3)]

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

lim_ (n rarr oo) [(1 ^ (3)) / (n ^ (4) + 1 ^ (4)) + (2 ^ (3)) / (n ^ (4) + 2 ^ (4)) ++ (1) / (2n)] =

S1: lim_(n->oo) (2^n + (-2)^n)/2^n does not exist S2: lim_(n->oo) (3^n + (-3)^n)/4^n does not exist

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

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

The value of lim_(n rarr oo) (1 + 2^(4) + 3^(4) +…...+n^(4))/(n^(5)) - lim_(n rarr oo) (1 + 2^(3) + 3^(3) +…...+n^(3))/(n^(5)) is :

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