Show that `2.6.10…….`to n factors `=((2n)!)/(n!)`

If n is a natural number then show that 1.3.5.7…..(2n-1) = ((2n)!)/(2^n n!)

the continued product 2.6.10.14... to n factors is equal to 2np_(n)(b)^(2n)C_(n)(n+1)(n+2)(n+3)...(n+n)2^(n)*(1.3.5......2n-1)

If there is a term independent of x in (x+(1)/(x^(2)))^(n) ,show that it is equal to (n!)/(((n)/(3))!((2n)/(3))!)

Show that, lim_(n to oo)((1)/(n + 1)+(1)/(n+2)+…+(1)/(6n))=log 6 .

Show that lim_(n rarr oo)((1)/(n+1)+(1)/(n+2)+...+(1)/(6n))=log6

Show tha , lim_(ntooo) ((1)/( n +1)+(1)/( n +2)+ . . . +(1)/( 6 n))= log 6 .

Some of the factors of (n ^(2))/(2) + (n)/(2) are (1)/(2) ,n and (n+1). True or false.

Show that the HM of (2n+1)C_(-) and (2n+1)C_(-)(r+1) is (2n+1)/(n+1) xx of (2n)C_(r) Also show that sum_(r=1)^(2n-1)(-1)^(r-1)*(r)/(2nC_(r))=(n)/(n+1)

Show that if n>2cos^(2)((pi)/(n))+cos^(2)((3 pi)/(n))+cos^(2)((5 pi)/(n))+...nterms=(n)/(2)

Show that n!(n+2)=n!+(n+1)!