  Inspired by the method of smoothed asymptotics developed by Terence Tao, we
introduce a new ultra-violet regularisation scheme for loop integrals in
quantum field theory which we call $\eta$ regularisation. This allows us to
reveal a surprising connection between the elimination of divergences in
divergent series of powers and the preservation of gauge invariance in the
regularisation of loop integrals in quantum field theory. In particular, we
note that a method for regularising the series of natural numbers so that it
converges to minus one twelfth inspires a regularisation scheme for non-abelian
gauge theories coupled to Dirac fermions that preserves the Ward identity for
the vacuum polarisation tensor. We also comment on a possible connection to
Schwinger proper time integrals.
