# estimation of sums $\sum_{1\le x \le n} [ f(x)]$

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