Prove that
`1+(1+x)+(1+x+x^(2))+(1+x+x^(2)+x^(3))+...+` to n terms `=(n)/(1-x)-(x(1-x^(n)))/(1-x)^(2)`

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

Prove that cos^(-1)((1-x^(2n))/(1+x^(2n)))=2tan^(-1)x^(n),0

Prove that 1-""^(n)C_(1) (1+x)/(1+n x)+""^(n)C_(2) (1+2x)/((1+n x)^(2)) ""^(n)C_(3) (1+3x)/((1+n x)^(3)) +….(n+1) terms = 0

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.

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.

Let f(x)=x+(1-x)x^(2)+(1-x)(1-x^(2))x^(3)+............+(1-x)(1-x^(2)).........(1-x^(n-1))x^(n);(n) then :

Prove that : If n is a positive integer and x is any nonzero real number, then prove that C_(0)+C_(1)(x)/(2)+C_(2).(x^(2))/(3)+C_(3).(x^(3))/(4)+….+C_(n).(x^(n))/(n+1)=((1+x)^(n+1)-1)/((n+1)x)

If y=1+(x)/(1!)+(x^(2))/(2!)+(x^(3))/(3!)+ . . .+(x^(n))/(n!) , prove that (dy)/(dx)+(x^(n))/(n!)=y

lim_(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^(2)), n in N , equals