1+2+3+...........+n<(1)/(8)(2n+1)^(2)

Lt_(ntooo)[(1)/(3n+1)+(1)/(3n+2)+............+(1)/(3n+n)]=

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 :

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

Lt_(ntooo)[(n^(1//2))/(n^(3//2))+(n^(1//2))/((n+3)^(3//2))+(n^(1//2))/((n+6)^(3//2))+.......+(n^(1//2))/({n+3(n-1)}^(3//2))]=

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

{:(" " Lt),(n rarroo):} [(1^(2))/(n^(3)+1^(3))+(2^(2))/(n^(3)+2^(3))+......+(1)/(2n)]=

lim_(nto 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)

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

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