- #26

Hurkyl

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Wrong. But I'll chalk that up to not knowing what I meant by "generalized function". If the functions [itex]f_n[/itex] are defined byGee I must've missed that bit..

NOT!

It IS valid for many sorts of generalized functions as is apparent from my 2nd post (though that's not explicitly Ramanujan summation, for more consider the Euler MacLaurin formula)

[tex]f_n(x) := \begin{cases} \frac{1}{2n} & x \in [-n, n] \\ 0 & x \notin [-n, x] \end{cases}[/tex]

Then the limit [itex]\lim_{n \rightarrow 0} f_n[/itex] doesn't exist. But if we instead work in the space of, for example, Schwartz distributions, then the limit exists and is equal to the delta function.

The general procedure here is to expand your universe of discourse so that (ordinary!) limits and sums that failed to converge in the old universe really do converge in the new universe.

This is an entirely different method than what you've been considering. You retain the

**ordinary**definition of "limit", "sum", "integral", et cetera, but you expand the class of objects you're working with.

Nobody said otherwise. The point is that they areThese new summation operators are most all legitimate, can be made rigorous, consistent

*summation operators, and the class of valid relations involving these*

**NEW***summation will be different than the class of valid relations involving the*

**NEW***summation operator.*

**OLD**Can the attitudeand are most inconveniently for those who curl up in their comfy math beds(or dare I say Couches) - true.

It is all in the definition as Hurkyl apparently concedes and does not invade upon the rigours of pure mathematics(as that is indeed my main area of interest)

*right now*.

Well duh. The summation operator from elementary calculus has a precise definition that distinguishes it from all other possible operators. The other (rigorous) generalizations of the summation operator I know also have precise definitions that distinguish them from all other possible operators.Oh and I noticed you seem to think of the summation operator as uniquely defined