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