despite that no real-world process of “addition” involving infinitely many terms may be performed in a finite number of steps, we can write
1+2+3+4+…=−112.
Well, not precisely. Because the first series converges, there’s a whole bunch more we can practically do with the equivalence-assignment in the first series, like using it as an approximation for the sum of any finite number of terms. −1/12 is a terrible approximation for any of the partial sums of the second series.
IMO the use of “=” is actually an abuse of notation by mathematicians in both cases above, but at least an intuitive / forgivable one in the first case because of the usefulness of approximating partial sums. Writing things as (1,2,3,...)∼−112 or R((1,2,3,...))=−112 (R() denoting Ramanujan summation, which for convergent series is equivalent to taking the limit of partial sums) would make this all less mysterious.
In other words, (1, 2, 3, …) is in an equivalence class with −1/12, an equivalence class which also contains any finite series which sum to −1/12, convergent infinite series whose limit of partial sums is −1/12, and divergent series whose Ramanujan sum is −1/12.
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.
Well, not precisely. Because the first series converges, there’s a whole bunch more we can practically do with the equivalence-assignment in the first series, like using it as an approximation for the sum of any finite number of terms. −1/12 is a terrible approximation for any of the partial sums of the second series.
IMO the use of “=” is actually an abuse of notation by mathematicians in both cases above, but at least an intuitive / forgivable one in the first case because of the usefulness of approximating partial sums. Writing things as (1,2,3,...)∼−112 or R((1,2,3,...))=−112 (R() denoting Ramanujan summation, which for convergent series is equivalent to taking the limit of partial sums) would make this all less mysterious.
In other words, (1, 2, 3, …) is in an equivalence class with −1/12, an equivalence class which also contains any finite series which sum to −1/12, convergent infinite series whose limit of partial sums is −1/12, and divergent series whose Ramanujan sum is −1/12.
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.