Name for this power sum formula?

Question

http://mathworld.wolfram.com/PowerSum.html

$$\sum_{k=0}^n kx^k = \frac{x-(n+1)x^{n+1}+nx^{n+2}}{(x-1)^2}$$

Answer 1

This is analogous to the derivative of the sum of a finite geometric series. (It is not precisely that, it requires a multiplication of the ratio, but it's as close a way to describe it as I know.) I don't know a precise name for the sum however.

Recall, for a finite geometric series,

$$\sum_{k=0}^n x^k = \frac{x^{n+1} - 1}{x-1}$$

Take the derivative of each side with respect to $x$; with some simplification,

$$\sum_{k=1}^n kx^{k-1} = \frac{1 - (n+1) x^n + n x^{n+1}}{(x-1)^2}$$

Multiply through by $x$ (thus achieving $x^k$ in the sum). Then

$$\sum_{k=1}^n kx^{k} =\frac{x - (n+1) x^{n+1} + n x^{n+2}}{(x-1)^2}$$

Note that while your sum has an $k=0$ term, that term is zero $(0x^0)$, and thus unnecessary; i.e. it is equivalent to the sum above.