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 x is positive, the sum to infinity of the series 1/(1+x)-(1-x)/((1+x)^2)+(1-x) ^2/((1+x)^3)-(1-x)^3/((1+x)^4)+....... i s (a)1/2 (b) 3/4 (c) 1 (d) none of these

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

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

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

Sum to n terms of the series (1)/((1 +x) (1 + 3x)) + (1)/( ( 1 + 3x) (1 + 5x)) + (1)/((1 + 5x ) (1 + 7x)) +…….

Find the sum of 20 terms of the series (x+(1)/(2))+(3x-(1)/(6))+(5x+(1)/(18))+....

Let n is a rational number and x is a real number such that |x|lt1, then (1+x)^(n)=1+nx+(n(n-1)x^(2))/(2!)+(n(n-1)(n-2))/(3!).x^(3)+ . . . This can be used to find the sm of different series. Q. The sum of the series 1+(1)/(3^(2))+(1*4)/(1*2)*(1)/(3^(4))+(1*4*7)/(1*2*3)*(1)/(3^(6))+ . . . . is

The sum of the series (1)/(1!)x+(2)/(2!)x^(2)+(3)/(3!)x^(3) ….. To infty is

The sum of the series (x-1)/(x+1)+1/2(x^(2)-1)/(x+1)^(2)+1/3(x^(3)-1)/(x+1)^(3) +…is equal to

Find the sum of the series 1+2^(2)x+3^(2)x^(2)+4^(2)x^(3)+"...."" upto "infty|x|lt1 .