If m is negative or positive and greater than 1 , show that `1^(m)+3^(m)+5^(m)+….+(2n-1)^(m) gt n^(m+1)`

If mgt1 and ninN , such that 1^(m)+2^(m)+3^(m)+...+n^(m)gtn((n+1)/(k))^(m) Then, k=

If m and n are positive quantites , prove that ((mn+1)/(m+1))^(m+1)ge n^(m)

If m >1,n in N show that 1^m+2^m+2^(2m)+2^(3m)++2^(n m-m)> n^(1-m)(2^n-1)^mdot

If x,y are positive real numbers and m, n are positive integers, then prove that (x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)

If a ,\ m ,\ n are positive integers, then {root(m)root (n)a}^(m n) is equal to (a) a^(m n) (b) a (c) a^(m/n) (d) 1

Statement-1: 1^(3)+3^(3)+5^(3)+7^(3)+...+(2n-1)^(3)ltn^(4),n in N Statement-2: If a_(1),a_(2),a_(3),…,a_(n) are n distinct positive real numbers and mgt1 , then (a_(1)^(m)+a_(2)^(m)+...+a_(n)^(m))/(n)gt((a_(1)+a_(2)+...+a_(b))/(n))^(m)

Show that if a and b are relatively prime positive integers, these there exist integers m and n such that a^(m)+b^(n) -=1 (mod ab).

lim_(n to oo){(1^(m)+2^(m)+3^(m)+...+ n^(m))/(n^(m+1))} equals

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4