A new perspective on Ramanujan summation: Invariant extraction

Assigning finite values to divergent series ∑an —such as 1+2+3+⋯=−112 —is well established, yet the conceptual meaning of such assignments remains elusive. We offer a perspective from digital signal processing: project (an) onto a measurement sequence and read the classical finite value as the scale-invariant component of that projection—the Sequence Invariant (SI). For projections admitting a Laurent expansion at the origin, the SI is invariant under scale (SCL), and also under shift (SFT) when the projection is regular there. The framework agrees with the Abel mean when it exists yet remains applicable when partial sums diverge to +∞ or −∞ rather than oscillate without bound, so the Abel mean has no finite value. It also recovers ζ(−k) and η(−k) and supplies constructive numerical algorithms.