Show that: `(x)+(x+1/n)+(x+2/n)+...+(x+(n-1)/n) = nx + (n-1)/2`

If x + y = 1, show that D ^ (n) (x ^ (n) y ^ (n)) = n! [Y ^ (n) - (nC_ (1)) ^ (2) y ^ (n -1) x + (nC_ (2)) ^ (2) y ^ (n-2) x ^ (2) + ...... + (- 1) ^ (n) x ^ (n)]

n + (n-1) x + (n-2) x ^ (2) + ....... + 2x ^ (n-2) + x ^ (n-1)

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

Show that If I _(n) =int cos ^(n) x dx, then show that I_(n) =1/n cos ^(n-1) x sin x + (n-1)/(n ) I_(n-2).

Show that 1+2x + 3x^2 +….+ nx^(n-1) = (1-(n+1)x^(n) + nx^(n+1))/((1-x)^2) for all n in N .

If (n^(n))/(n!)=(nx)/((n-1)!), x =

Show that x^(n) =1 + n(1 -1/x) + (n(n+1))/1.2 (1 -1/x)^(2) + ...

Show that 1/(x+1)+2/(x^2+1)+4/x^4+1)+…..+2^n/(x^2^n+1)= /(x-1)- 2^(n+1)/(x^2(n+1) -1)

Show that 1/(x+1)+5/(x^2+1)+4/x^4+1)+…..+2^n/(x^2^n+1)= /(x-1)- 2^(n+1)/(x^2(n+1) -1)

Show that (n !)/(x(x+1)(x+2)...(x+n))=(""^(n)C_(0))/(x)-(""^(n)C_(1))/(x+1) + (""^(n)C_(2))/(x+2)-(""^(n)C_(3))/(x+3) + ...+ ((-1)^(n)""^(n)C_(n))/(x+n) .