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Please does someone know a good description of how Euler did derive his summation formula? Thank you!

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See also the references in–Maclaurin_formula – Dietrich Burde Apr 27 '14 at 11:44

Most of Euler's work is available at "The Euler Archive".

The papers of interest in this case (according to Ferraro's "Euler's derivation of the Euler–Maclaurin summation formula") are :

Further references :

Ferraro's article "Some Aspects of Euler's Theory of Series - Inexplicable Functions and the Euler–Maclaurin Summation Formula" details Euler's different derivations.
Hairer and Wanner's book "Analysis by its History" provides a short Euler derivation.
Pengelley's "Dances between continuous and discrete" considers Euler's derivations too.

Apostol's "An Elementary View of Euler's Summation Formula" may clarify things.
Lampret's derivation may be of interest too.

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