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The series 1-1+1-... does not converge to anything in particular, but is "summable" to 1/2 [G.H.Hardy, "Divergent series", pp.6-7]. That is to say, there exists an axiomatic framework within which the assignment of 1/2 as the sum of the series 1+1-1+1-... is unambiguous. In fact, Hardy goes on to call the (really) simple axiomatic framework (two axioms of manipulation of series—and, most importantly, which are perfectly valid for convergent series—suffice to compute a value for this series) "Pickwickian." So, it is most certainly not merely a "zeta-funciton complex analysis

trick" that makes 1-1+1-... = 1/2. The point (implied, but perhaps not sufficiently emphasized by your differentiating between Euler's and Riemann's zeta function!) is that since a series such as "1-1+1-..." does not converge to anything in particular, it requires a context (an axiomatic framework) within which to acquire a meaningful value.