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I'd like to approximate a sum of the form $ S(n)=\sum\limits_{k=1}^{n}\phi\left(\frac{k}{n}\right)$ with an integral using Riemannian sums:

$$S(n) \approx n \int_{0}^{1}\phi(x)dx +o(n).$$

My concern is, $k$ in the argument can't be less than $1$. Should I bound the integral between $[0,1]$ or $[1,2]$?

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Did you mean to write $S(n) = \sum \limits_{k=1}^n \phi(k/n)$? –  Srivatsan Jan 2 '12 at 3:22
    
yes, you are right.corrected. –  user19821 Jan 4 '12 at 1:34
    
Would you say that the set of k/n from k=1 to n acts more as a representative sample from the interval [0,1] or from the interval [1,2]? –  Did Jan 5 '12 at 0:51
    
@DidierPiau:I'm not sure I understood your question. The reason I want to do it is because $k<1$ is not sensible. –  user19821 Jan 5 '12 at 4:52
    
For $1\leqslant k\leqslant n$, would you say $\phi(k/n)$ is a value that the function $\phi$ assumes at (i) a point of the interval $[0,1]$, or (ii) a point of the interval $[1,2]$? –  Did Feb 25 '12 at 0:57

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