I write "may" since many books have different conventions for representing the Bernoulli polynomials, but having reviewed the conventions that Fungrim follows, I think the sign of the remainder term needs to be negative. I am referring to Asymptotic Methods in Analysis by N. G. be Bruijn which states the Euler-Maclaurin summation formula in almost exactly the same form, but with a negative sign on the remainder. Having reviewed the integration by parts derivation of the formula very closely, I believe the book is correct and Fungrim is probably in error.
When doing the integration by parts, the signs alternate on the remainder term, but since Fungrim always refers to the 2m'th Bernoulli polynomial, the sign should always be negative I believe.
Hi,
I write "may" since many books have different conventions for representing the Bernoulli polynomials, but having reviewed the conventions that Fungrim follows, I think the sign of the remainder term needs to be negative. I am referring to Asymptotic Methods in Analysis by N. G. be Bruijn which states the Euler-Maclaurin summation formula in almost exactly the same form, but with a negative sign on the remainder. Having reviewed the integration by parts derivation of the formula very closely, I believe the book is correct and Fungrim is probably in error.
When doing the integration by parts, the signs alternate on the remainder term, but since Fungrim always refers to the
2m'th Bernoulli polynomial, the sign should always be negative I believe.