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.
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.