Add new comment
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
A game you're almost certain to lose...
What are the challenges of communicating from the frontiers of mathematical research, and why should we be doing it?
Celebrate Pi Day with the stars of our podcast, Maths on the move!
Maths meets politics as early career mathematicians present their work at the Houses of Parliament.
Celebrate this year's International Women's Day with some of the articles and podcasts we have produced with women mathematicians over the last year!
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.