What is the sum of the series $x^{n-1} + 2x^{n-2} + 3x^{n-3} + 4x^{n-4} + \cdots + nx^{n-n}$? $x^{n-1} + 2x^{n-2} + 3x^{n-3} + 4x^{n-4} + \cdots + nx^{n-n}$
I'm not sure if this can even be summed. Any help is appreciated. 

It is
$S=x^{n-1}+2x^{n-2}+3x^{n-3}+\ldots+(n-2)\cdot x^2+(n-1)\cdot x^1+n\cdot x^0 \quad (1)$
$x\cdot S=x^{n}+2x^{n-1}+3x^{n-2}+\ldots+(n-2)\cdot x^3+(n-1)\cdot x^2+n\cdot x^1 \quad (2)$
Subtracting (1) from (2):
$(x-1)S=\color{blue}{x^n+x^{n-1}+x^{n-2}+x^{n-3}+\ldots + x^2+x}-n$
The formula for the blue expression is $x\cdot \frac{1-x^{n}}{1-x}$ (geometric series)
Now solve for $S$.
