<s>[INST] 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. 
 [/INST] \begin{align}
& x^{n-1} + 2x^{n-2} + 3x^{n-3} + 4x^{n-4} + \cdots + nx^{n-n} \\[10pt]
& x^{n-1}\left( 1 + \frac 2 x + \frac 3 {x^2} + \frac 4 {x^3} + \cdots + \frac n {x^{n-1}} \right) \\[10pt]
= {} & x^{n-1}\left( 1 + 2y + 3y^2 + 4y^3 + \cdots + ny^{n-1} \right) & (\text{where }y = 1/x) \\[10pt]
= {} & x^{n-1} \frac d {dy} \left( 1 + y + y^2 + y^3 + y^4 + \cdots + y^n \right) \\[10pt]
= {} & x^{n-1} \frac d {dy} \, \frac{1 - y^{n+1}}{1-y}.
\end{align}
Now do the differentiation and then finally put $1/x$ wherever $y$ appears.
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