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