Alternating Ordinary Generating Functions

SECTIONS

Alternate Geometric Series

$\displaystyle \dfrac{a}{1+x} = \sum_{n \geq 0} a \, (-1)^n \, x^n \\ = a - ax + ax^2 - ax^3 + ax^4 - ax^5 \pm \cdots \$

$\displaystyle \dfrac{1}{1+ax} = \sum_{n \geq 0} (-a)^n \, x^n \\ = 1 - ax + (ax)^2 - (ax)^3 + (ax)^4 - (ax)^5 \pm \cdots \$

Alternate Polygonal Series

$\displaystyle \dfrac{1-(r-3)x}{(1+x)^3} = \sum_{n \geq 1} (-1)^{n-1} P_r(n) x^n \\ = P_r(1) - P_r(2) + P_r(3) - P_r(4) + P_r(5) - P_r(6) \pm \cdots$

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