`AAn in N ,1+2x+3x^2++ndotx^(n-1)=(x in R , x!=1)`

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

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) ] =

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) .

Find the sum 1+2x+3x^2+…..+nx^(n-1) from the relation 1+x+x^2+x^3+…..+x^n=(1-x^(n+1))/(1-x)

AA 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))]=

For x in R -{-1/n, n in N} , define f(x) = lim_(n to infty)(x/(x+1) + x/((x+1)(2x+1)) + x/((2x+1)(3x+1))) + ….. + upto n terms then range of f contains exactly……….. Element(s).

If A=([x,x],[x,x]) then A^(n)(n in N)= 1) ([2^nx^n,2^nx^n],[2^nx^n,2^nx^n]) 2) ([2^(n-1) x^n,2^(n-1) x^n],[2^(n-1) x^n,2^(n-1) x^n]) 3) I 4) ([2^(n) x^(n-1),2^(n) x^(n-1)],[2^(n) x^(n-1),2^(n) x^(n-1)])

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) .