`Limn→∞(1+2+3...........n)/(2n^2)`

Prove that ((2n+1)!)/(n !)=2^n{1. 3. 5 .........(2n-1)(2n+1)}

The value of lim_(n to oo)((1)/(1^(3)+n^(3))+(2^(2))/(2^(3)+n^(3))+..........+(n^(2))/(n^(3)+n^(3))) is :

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

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

Statement-1: 1+2+3....+n=(n(n+1))/(2),"for all "n in N Statement-2: a+(a+d)+(a+2d)+....+(a+(n-1)d)=(n)/(2)[2a+(n-1)d]

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

f(n) = cot^2 (pi/n) + cot^2\ (2 pi)/n +...............+ cot^2\ ((n-1) pi)/n, ( n>1, n in N) then lim_(n rarr oo) f(n)/n^2 is equal to (A) 1/2 (B) 1/3 (C) 2/3 (D) 1

If S_n=1/1^3 +(1+2)/(1^3+2^3)+...+(1+2+3+...+n)/(1^3+2^3+3^3+...+n^3) Then S_n is not greater than

(1^4)/1.3+(2^4)/3.5+(3^4)/5.7+......+n^4/((2n-1)(2n+1))=(n(4n^2+6n+5))/48+n/(16(2n+1)

The value of lim_(n->oo) (1^2 . n+2^2.(n-1)+......+n^2 . 1)/(1^3+2^3+......+n^3) is equal to