sumninftyinfty

Sumninftyinfty is a fictional mathematical operation introduced in online puzzle and science-fiction communities to explore how divergent series might be assigned finite values under controlled regularization rules. It is not part of formal mathematics but serves as a thought experiment for comparing different summation methods and for world-building in mathematical fiction.

In toy formulations, given a sequence a_k with partial sums s_N = sum_{k=1}^N a_k, the sumninftyinfty of the sequence is defined by S = lim_{N→∞} F_N(s_N), where F_N is a normalization function chosen according to a predefined scheme. Depending on the scheme, S agrees with ordinary sums for convergent series, while divergent sums receive finite values under regularization. Common toy schemes include symmetric cutoff and analytic-continuation-inspired rules.

Example: For the alternating series a_k = (-1)^{k+1}, a Cesàro-type sumninftyinfty assigns the value 1/2.

Origin and note: The term is fictional and used mainly in collaborative wikis and puzzle communities to illustrate regularization concepts; it has no official status in real analysis. Related ideas include Cesàro and Ramanujan summation, zeta-function regularization, and other divergent-series techniques.