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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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Since the floor function is constant where it is differentiable, any power series representation will be a constant. –  Grumpy Parsnip Mar 14 '12 at 12:49
    
Riccardo.Alestra has given a formula when the value is not an integer. Is there a known formula that includes integer values also? –  Matt Groff Mar 14 '12 at 12:54
    
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. –  Grumpy Parsnip Mar 14 '12 at 12:59
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1 Answer

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

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

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