`1/(x+1)+1/(x+1)^2+1/(x+1)^3+..................`

Let f(x)=x+(1-x)x^3+(1-x)(1-x^2)x^3+.........+(1-x)(1-x^2)........(1-x^(n-1))x^n;(n >= 4) then :

(1)/(x+1)+(1)/(2(x+1)^(2))+(1)/(3(x+1)^(3))+.... =

Find the sum of the series 1+2(1-x)+3(1-x)(1-2x)+....+n(1-x)(1-2x) (1-3x)............[1-(n-1)x].

Find the sum of the series 1+2(1-x)+3(1-x)(1-2x)+....+n(1-x)(1-2x) (1-3x)............[1-(n-1)x].

If f(x)=(1)/(x+1)+(1)/(2(x+1)^(2))+(1)/(3(x+1)^(3))+…(x gt 1) and f(1), f(2), f(3) are respectively p, q, r then their ascending order is

If f(x)=(1)/(x+1)+(1)/(2(x+1)^(2))+(1)/(3(x+1)^(3))+…(x gt 1) and f(1), f(2), f(3) are respectively p, q, r then their ascending order is

(x)/(x+1)+(1)/(2)((x)/(x+1))^(2)+(1)/(3)((x)/(x+1))^(3)+....=

(x)/(x+1)+(1)/(2)((x)/(x+1))^(2)+(1)/(3)((x)/(x+1))^(3)+....=

Without expanding, find the value of: (i) (x + 1)^4 - 4(x + 1)^3 (x - 1) + 6(x + 1)^2 (x - 1)^2 - 4(x + 1) (x - 1)^3 + (x -1)^4 (ii) (2x - 1)^4 + 4(2x - 1)^3 (3 - 2x) + 6(2x - 1)^2 (3 - 2x)^2 + 4(2x - 1) (3 - 2x)^3 + (3 - 2x)^4