  A route to evaluate exact sums represented by Dirichlet eta and beta
functions, both of which are alternating and divergent at negative integer
arguments, is advocated. It rests on precise polynomial extrapolations and
stands as a generalization of an early endeavor on lattice sums. Apart from
conferring a physical meaning to anti-limits, the scheme advanced here is
direct, independent and computationally appealing. A new interpretation of
summability is also gained.
