Which form of Euler-Maclaurin formula to use?

Question

This question may be rather elementary, but I am sort of confused about various forms of the Euler-Maclaurin summation formula and their use.

For instance, let us suppose that we want to approximate the sum $$\sum_{k=0}^n \sin\frac{k\pi}{n}$$ by the corresponding integral $$\int_0^{\pi}\sin x\,\mathrm{d}x.$$ Which form of Euler-Maclaurin formula is the best choice to use now? On the one hand, there is a form of the Euler-Maclaurin formula that allows us to approximate the sum by the (clearly equivalent) integral $$\int_0^n \sin \frac{x\pi}{n}\,\mathrm{d}x.$$ I believe that this is possible, but is this the most efficient way of doing so? Although the integral itself can be clearly transformed to the simple integral above, the computation of the reminder terms may not be so simple.

On the other hand, there is a form of the Euler-Maclaurin formula that is intended for approximating sums on the fixed interval. However, the step size is considered to be fixed here.

So my question essentially is: which form of the Euler-Maclaurin formula would you use to approximate sums of the above type: that is, essentially, sums where we sum according to some set of points in a fixed interval, where the set becomes denser and denser.

My apologies, if this question does not make much sense.

Answer 1

I would use Euler summation for $\sum_{k=0}^n f(k)$ with $f(k)=\sin(kx)$, and $\int_0^n f(x)dx$. How efficient this is, depends on the error term. There is also a formula for the above sum, i.e., $$ \sum_{k=1}^n\sin(kx)=\frac{\cos \frac{x}{2}-\cos \frac{(n+1)x}{2}}{2\sin \frac{x}{2}}. $$