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)`

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.

Show that (x-1) is a factor of x ^(n)-1.

Using mathematical induction , show that (1-(1)/(2^2))(1-(2)/(3^2))(1-(1)/(4^2)).....(1-(1)/((n+1)^2))=(n+2)/(2(n+1)), forall n in N .

Find the sum of (x + (1)/(x))^(2), (x^(2) + (1)/(x^(2)))^(2), (x^(3) + (1)/(x^(3)))^(2) , ……..to n terms

Prove that 1^2/(1.3)+2^2/(3.5)+3^2/(5.7)+.....+n^2/((2n-1)(2n+1))=((n)(n+1))/((2(2n+1))

For all ngt=1 , prove that , (1)/(1.2) + (1)/(2.3) + (1)/(3.4) + ……+ (1)/(n(n+1)) = (n)/(n+1)

Using the principle of mathematical induction to show that tan^(−1)(x/(1+1.2.x^2))+tan^(−1)(x/(1+2.3.x^2))+.....+tan^(−1)(x/(1+n(n+1)x^2)) = tan^(-1)(n+1)x-tan^(-1)x , forall x in N .

Show that, (1+5^(-1))(1+5^(-2))(1+5^(-4))(1+5^(-8))"....."(1+5^(-2n))=(5)/(4)(1-5^(-2(n+1))) .

x-y relation is given by y=x^(n) calculate the value of n in the following :- (i) y=(x^(2))/(x^(2//3)) (ii) y=(x^(3//2))/(x^(1)) (iii) y=(x^(1//2))/(x^(1//2)) (iv) y=(x^(3//2))/(sqrt(x))

lim_(xto1)(x+x^(2)+x^(3)+…….+n^(n)-n)/(x-1) =……….,where n in N