(2+4x+6x^2+8x^3+\cdots+(2n)x^{n-1})

[ S=2\sum_{k=1}^{n} k x^{k-1} ] Or (S=2+4x+6x^2+\cdots+2n x^{n-1})

Standard result: [ S=\dfrac{2\bigl[1-(n+1)x^n+nx^{n+1}\bigr]}{(1-x)^2} ]

Answer: (\dfrac{2[1-(n+1)x^n+nx^{n+1}]}{(1-x)^2})