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I would like to pay attention to an article " Sums and Integrals: The Swiss analysis knife " by Bill Casselman, where the Euler-Maclaurin formula is discussed.  It should be noted all that matter is implemented in Maple through the commands bernoulli and eulermac. For example,

bernoulli(666);


eulermac(1/x, x);

,

eulermac(sin(x), x, 6);

BTW, as far as I know it, this boring stuff is substantially used in modern physics. The one is considered in

Ronald Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, Addison-Wesley, 1989.

The last chapter is concerned with the Euler-MacLaurin formula.


           
integral sum bernoulli euler-maclaurin

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