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

Prove by mathematical induction that (1)/(1+x)+(2)/(1+x^2)+(4)/(1+x^4)+.....+(2^n)/(1+x^(2^n))=(1)/(x-1)+(2^(n+1))/(1-x^(2^(n+1))) where , |x|ne 1 and n is non - negative integer.

cos^(-1)((1-x^(2n))/(1+x^(2n)))

Obtain the sum of (1)/(x+1)+(2)/(x^(2)+1)+(4)/(x^(4)+1)+......+(2^(n))/(x^(2^(n))+1)

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

lim/(x to2)(sum^(9)/(n =1) x/(n(n +1) x^2 +2(2n +1)x + 4))

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

Prove that int(dx)/((1+x^(2))^(n))=(1)/(2(n-1))[(x)/((1+x^(2))^(n-1))+(2n-3)int(dx)/((1+x^(2))^(n-1))],n in N Hence,computer the value of int cos^(4)xdx

1+x+(1)/(2!)x^(2)+...+(1)/((2n)!)x^(2n)=0