Prove that `D^n((logx)/x)=(-1)^n n ! x^(-n-1)[logx-1-1/2-1/3-.......1/n]`

Prove that cos^(-1)((1-x^(2n))/(1+x^(2n)))=2 tan^(-1)x^n

Prove that : int_(0)^(1)x(1-x)^(n)dx=1/((n+1)(n+2))

Prove that ((2n+1)!)/(n!)=2^(n){1.3.5(2n-1)(2n+1)}

If A=[[1 , 1, 1],[ 1, 1, 1],[ 1, 1 , 1]] , prove that A^n=[[3^(n-1), 3^(n-1) , 3^(n-1)],[ 3^(n-1), 3^(n-1) , 3^(n-1)],[ 3^(n-1) , 3^(n-1), 3^(n-1)]] n in N .

Prove that ((2n+1)!)/(n!)=2^(n)[1.3.5.....(2n-1)*(2n+1)]

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

If A=[[1, 1, 1],[ 1, 1, 1],[ 1, 1, 1]] , prove that A^n=[[3^(n-1),3^(n-1),3^(n-1)],[3^(n-1),3^(n-1),3^(n-1)],[3^(n-1),3^(n-1),3^(n-1)]], n in N.

If A=[1 1 1 1 1 1 1 1 1] , prove that A^n=[3^(n-1)3^(n-1)3^(n-1)3^(n-1)3^(n-1)3^(n-1)3^(n-1)3^(n-1)3^(n-1)], n in Ndot

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

If A = [[1,1,1],[1,1,1],[1,1,1]] , prove that A^n = [[3^(n-1),3^(n-1),3^(n-1)],[3^(n-1),3^(n-1),3^(n-1)],[3^(n-1),3^(n-1),3^(n-1)]], n in N