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I am interested in the way Mathematica evaluates sums where the upper limit of the index is not an integer - i.e., sums that are some sense mathematically meaningless. For example:

{Sum[Cos[x - i], {i, 0, x}], Sum[Cos[x - i], {i, 0, Floor[x]}]} /. x -> 3.5

(*
 {-0.2892805393, -0.7892805393}
*)

The second number is clear and accurate. But what does that first number actually represent?

What assumptions has Mathematica made in order to be able to evaluate the sum?

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Sum[foo[i], {i, 1, 3.5}]
foo[1] + foo[2] + foo[3]

How -0.289281 is obtained in the first case in OP: Sum evaluates before ReplaceAll takes effect:

Sum[Cos[x - i], {i, 0, x}]
1/2 (1 + Cos[x] + Cot[1/2] Sin[x])
% /. x -> 3.5
-0.289281

You can see this using Trace:

Trace[Sum[Cos[x - i], {i, 0, x}] /. x -> 3.5] // Column

enter image description here

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