AA n in N,1+2x+3x^(2)+....+n*x^(n-1)=(x in R,x!=1)

AA n in , 1+2x + 3x^(2) + ….+ n.x^(n-1) = (x in R, x ne 1)

AA n in N, 1 + 2x + 3x ^ (2) ++ ndot x ^ (n-1) = (x in R, x! = 1)

From the relation 1+x+x^(2)+* * * + x^(n-1)=(1-x^(n))/(1-x) , find the sum of the series 1+2x+3x^(2)+* * * +(n-1)x^(n-2) .

n + (n-1) x + (n-2) x ^ (2) + ....... + 2x ^ (n-2) + x ^ (n-1)

Show that 1+2x + 3x^2 +….+ nx^(n-1) = (1-(n+1)x^(n) + nx^(n+1))/((1-x)^2) for all n in N .

Statement-1: The middle term of (x+(1)/(x))^(2n) can exceed ((2n)^(n))/(n!) for some value of x. Statement-2: The coefficient of x^(n) in the expansion of (1-2x+3x^(2)-4x^(3)+ . . .)^(-n) is (1*3*5 . . .(2n-1))/(n!)*2^(n) . Statement-3: The coefficient of x^(5) in (1+2x+3x^(2)+ . . .)^(-3//2) is 2.1.

Statement-1: The middle term of (x+(1)/(x))^(2n) can exceed ((2n)^(n))/(n!) for some value of x. Statement-2: The coefficient of x^(n) in the expansion of (1-2x+3x^(2)-4x^(3)+ . . .)^(-n) is (1*3*5 . . .(2n-1))/(n!)*2^(n) . Statement-3: The coefficient of x^(5) in (1+2x+3x^(2)+ . . .)^(-3//2) is 2.1.

For all n in N, x in R, tan^(-1) [ ( x)/( 1.2+ x^2) ] + tan^(-1) [ (x)/( 2.3+ x^2) ] + …. + tan^(-1) [ ( x)/( n(n+1) +x^2) ] =

It is known for n ne 1 that : 1+x+x^2+..........+x^(n-1)=(1-x^(n))/(1-x) , hence find the sum of the series: 1+2x+3x^(2)+"….."+(n-1)x^(n-2) .