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given a sum in the form $ \sum_{1\le x \le n} [ f(x)] =S $ , here $ [ x]$ is the floor function which takes only integer values.

how could i evaluate or give a good estimation ??

i know that the sum $ \sum_{1\le x \le n} f(x) - [ f(x)] $ would be less or equal than $ n-1 $ but what a better estimation can be done ?

could i expand $ [ f(x)] $ into a fourier series on the interval $ (1,n) $ to make it easier to evalute the series ?

wher could i find more references :D ?? thanks.

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Do you have a particular $f$? –  Berci Sep 28 '12 at 16:16
no, let us assume $ f(x) $ is smooth enough –  Jose Garcia Sep 28 '12 at 16:18
If $f$ is the smooth function $0$, then the error is $0$. If $f$ is the smooth function $1-\varepsilon$, your error is arbitrarily close to $n$. –  Hagen von Eitzen Sep 28 '12 at 16:21

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