an abuse of notation by mathematicians in both cases above
Writing things as (1,2,3,...)∼−112 or R((1,2,3,...))=−112
The point I was trying to make is that we already have perfectly good notation for sums, namely the + and = signs, that we’ve already extended well beyond the (apocryphal) original use of adding finite sets of positive integers. As long as there’s no conflict in meaning (where saying “there’s no answer” or “it’s divergent” doesn’t count) extending it further is fine.
My point is that there is a conflict for divergent series though, which is why 1 + 2 + 3 + … = −1/12 is confusing in the first place. People (wrongly) expect the extension of + and = to infinite series to imply stuff about approximations of partial sums and limits even when the series diverges.
My own suggestion for clearing up this confusion is that we should actually use less overloaded / extended notation even for convergent sums, e.g.(1,12,14,...)∼2 seems just as readable as the usual lim→ and +...+ notation.
The point I was trying to make is that we already have perfectly good notation for sums, namely the
+and=signs, that we’ve already extended well beyond the (apocryphal) original use of adding finite sets of positive integers. As long as there’s no conflict in meaning (where saying “there’s no answer” or “it’s divergent” doesn’t count) extending it further is fine.My point is that there is a conflict for divergent series though, which is why 1 + 2 + 3 + … = −1/12 is confusing in the first place. People (wrongly) expect the extension of + and = to infinite series to imply stuff about approximations of partial sums and limits even when the series diverges.
My own suggestion for clearing up this confusion is that we should actually use less overloaded / extended notation even for convergent sums, e.g.(1,12,14,...)∼2 seems just as readable as the usual lim→ and +...+ notation.