Monotonic Ordinary Generating Functions

SUPERSPECIES

Geometric Series

$\displaystyle \dfrac{x}{1-x} = \sum_{n \geq 1} x^n = \sum_{n \geq 0} x^{n+1} = x + x^2 + x^3 + x^4 + x^5 + x^6 + \cdots \$

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

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

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

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