Alternating Ordinary Generating Functions

SUPERSPECIES

Alternate Geometric Series

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

$\displaystyle \dfrac{1}{1+x^2} = \sum_{n \geq 0} (-1)^n \, x^{2n} = 1 - x^2 + x^4 - x^6 + x^8 - x^{10} \pm \cdots \$

$\displaystyle \dfrac{x}{1+x^2} = \sum_{n \geq 0} (-1)^n \, x^{2n+1} = x - x^3 + x^5 - x^7 + x^9 - x^{11} \pm \cdots \$

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

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