`1+2+3+...+n<((n+2)^8/8),n in N`, is true for

lim_ (n rarr oo) ((1) / (2) +1+ (3) / (2) + ... + (n) / (2)) / (25n ^ (2) + n + 3)

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

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

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

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

(1^(3)+2^(3)+...+n^(3))/(1+3+5+...+(2n-1))=((n+1)^( 2))/(4)

Find the sum of the series: 1. n+2.(n-1)+3.(n-2)++(n-1). 2+n .1.

Find the sum of the series: 1. n+2.(n-1)+3.(n-2)++(n-1). 2+n .1.

Let S_(n) = ( 1)/( 1^(3)) + ( 1+2)/( 1^(3) + 2^(3)) +"...." + ( 1+ 2 + "...." + n)/(1^(3) +2^(3)"...."+n^(3)), n = 1,2,3,"....." , Then S_(n) is not greater than :