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

Definite integration as the limit of a sum : lim_(ntooo)(1^(p)+2^(p)+3^(p)+.......+n^(p))/(n^(p+1))=...........

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

The value of lim_(n to oo) (1^(p)+2^(p)+3^(p)+…+n^(p))/(n^(p+1)) is -

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

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) Prove that lim_ (n rarr oo) P_ ( n) = (2) / (3)

Let lim _( x to oo) n ^((1)/(2 )(1+(1 )/(n))). (1 ^(1) . 2 ^(2) . 3 ^(3)....n ^(n ))^((1)/(n ^(2)))=e^((-p)/(q)) where p and q are relative prime positive integers. Find the value of |p+q|.