" (iii) "(n+1)(n+2)(n+3)dots(2n)

Show that the middle term in the expansion of (1+x)^(2n) is (1.32n-1)/(n!)2^(n)dot x^(n)

(n!) / ((nr)!) = n (n-1) (n-2) dots (n- (r-1))

Evaluate(with the help of definite integral): lim_(n rarr oo){(1+(1)/(n))(1+(2)/(n))dots(1+(n)/(n))}^((1)/(n))

If n be a positive integer such that n<=3, then the value of the sum to n terms of the series 1.n-((n-1))/(1!)(n-1)+((n-1)(n-2))/(2!)(n-2)-((n-1)(n-2)(n-3))/(3!)(n-3)+dots

If n is a positive integer, prove that 1-2n+(2n(2n-1))/(2!)-(2n(2n-1)(2n-2))/(3!)+.......+(-1)^(n-1)(2n(2n-1)(n+2))/((n-1)!)= (-1)^(n+1)(2n)!//2(n !)^2dot

If n is a positive integer, prove that 1-2n+(2n(2n-1))/(2!)-(2n(2n-1)(2n-2))/(3!)++(-1)^(n-1)(2n(2n-1)(n+2))/((n-1)!)= (-1)^(n+1)(2n)!//2(n !)^2dot

Prove that (n!)/(r!)=n(n-1)(n-2)dots(r+1)

lim_(n rarr infty ) [((n+1)(n+2)...3n)/(n^(2n))]^(1//n) is equal to

lim_ (n rarr oo n rarr oo n pto n terms) (n) / ((n + 1) sqrt (2n + 1)) + (n) / ((n + 2) sqrt (2 (2n + 2)) ) + (n) / ((n + 3) sqrt (3 (2n + 3)) + dots)