Kurt,

I'm currently reading 
Zima, Eugene V.
Accelerating Indefinite Summation: Simple Classes of Summands"

which "presents the history of indefinite summation starting with
classics (Newton, Montmort, Taylor, Stirling, Euler, Boole, Jordan)
followed by modern classics (Abramov, Gosper, Karr) to the current
implementation in computer algebra system Maple".

It gives a nice overview up to but not including Schneider's work.

Tim



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