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I'm wondering if an ordinary power series expansion of the floor function is possible, and what it is if it's possible. Additionally, I'm wondering what some of the most common uses of the floor function are in practice. Additionally, I'd like to know what use a series expansion of a floor function may be.

So, again, what's the series expansion of the floor function?

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    Riccardo's formula is not a power series. I was only responding to the "power series" part of your question, not the more general question of whether there is any series expansion.2012-03-14

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If $x$ is not an integer, you can use the formula:

$\lfloor x \rfloor=x-\frac{1}{2}+\frac{1}{\pi}\sum_{k=1}^\infty \frac{\sin(2k\pi x)}{k}$