>>12809554Ramanujan summation is a unique definition to quantify how strongly divergent a series is. For example, see pic rel. Its value is negative, and magnitude is larger.
Theorem: If a series is Ramanujan summable, and its sum is bounded by 0 < S < infty, then it's a convergent series and its sum is equal to its Ramanujan sum. Proof: left as an exercise for the reader.
Corollary: If a Ramanujan series sums to infinity, it is divergent toward infinity. Proof: trivial.
Lemma: If a Ramanujan series sum to a negative value, then the magnitude of the Ramanujan sum represents the rate at which the series diverges to infinity. Proof: exercise.
See, for example pic rel.
1 + 2 + 4 + 8 + .... > 1 + 2 + 3 + 4 + ...
Despite the fact both diverge to infinity, the first series has a Ramanujan sum of -1, indicating more rapid divergence than the latter series which has a Ramanujan sum of -1/12.
