I think it's fair to say that summing the series directly would be slow, even if it's not slow when you already happen to have summed the previous n-1 terms.

Not least because for modestly-sized target sums the number of terms you need to sum is more than is actually feasible. For instance, if you're interested in approximating a sum of 100 then you need something on the order of exp(100) or about 10^43 terms. You can't just say "well, it's not slow to add up 10^43 numbers, because it's quick if you've already done the first 10^43-1 of them".