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+3x+5x^2+7x^3+...... n terms.

Find the sum of the series 1+2x+3x^2+(n-1)x^(n-2) using differentiation.

The sum of first n terms of the series 1 + (1 + x) y+ (1 + x + x^(2)) y^(2) + (1 + x + x^(2) + x^(3))y^(3) + ……….. is

Find the sum to ininity of the series 1-3x+5x^2+7x^3+..... oo when Ix| <1.

(3x+1)/((x-2)(x+1))

Let tan^(-1)y=tan^(-1)x+tan^(-1)((2x)/(1-x^2)) , where |x|<1/(sqrt(3)) . Then a value of y is : (1) (3x-x^3)/(1-3x^2) (2) (3x+x^3)/(1-3x^2) (3) (3x-x^3)/(1+3x^2) (4) (3x+x^3)/(1+3x^2)

If x in (0, 1) , then find the value of tan^(-1) ((1 -x^(2))/(2x)) + cos^(-1) ((1 -x^(2))/(1 + x^(2)))

Prove that the coefficient of x^n in the expansion of 1/((1-x)(1-2x)(1-3x))i s1/2(3^(n+2)-2^(n+3)+1)dot

Find the sum (x+2)^(n-1)+(x+2)^(n-2)(x+1)^+(x+2)^(n-3)(x+1)^2++(x+1)^(n-1)