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

Let S_(n)=1+(1)/(2)+(1)/(3)+….+(1)/(n) Show that ns_(n)=n+((n-1)/(1)+(n-2)/(2)+………+(2)/(n-2)+(1)/(n-1)) .

If n is a natural number then show that n! + (n+1)! = (n+2)n!

Show that ((n+1)!)/((n-2)!)=n^(3)-n3

If y=x^(2)e^(x) ,show that y_(n)=(1)/(2)n(n-1)y_(2)-n(n-2)y_(1)+(1)/(2)(n-1)(n-2)}

Show that ""^(n)P_(n)=2""^(n)P_(n-2)

Show that (n-1)/(n+1)+3((n-1)/(n+1))^2+5((n-1)/(n+1))^3+....+oo=sum_(r=1)^(n-1) r .

Let S_(n)=1+(1)/(2)+(1)/(3)+….+(1)/(n) Show that s_(n)=n-((1)/(2)+(2)/(3))+…….+(n-1)/(n)) .

Show that ""^(n)P_(n)=""^(n)P_(n-1) for all natural numbers n.

Show that (d^n)/(dx^(n) )(x^(n) log x) = n! (log x + 1+(1)/(2) +…+(1)/(n)) AA n in N .

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