10. T. E. Phipps Jr., Divergence: What to Do till the Mathematician Comes


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Volume 14: Pages 341-353, 2001

Divergence: What to Do till the Mathematician Comes

T. E. Phipps Jr.

908 South Busey Avenue, Urbana, Illinois 61801 U.S.A

It is argued that definitions may be chosen a posteriori to reflect mathematical existence—the opposite of the usual approach by which definitions are chosen a priori and used to prove existence. This inverted view is applied to the question of “divergence.” It is shown that a definition employing approximations to process remainders known as “terminal summation” allows “values” of divergent series to exist.

Keywords: divergence, divergent series, mathematics, definitions, convergence, infinite series, summability, terminal summation, Brown's function, Brown's series, Euler

Received: January 18, 2002; Published online: December 15, 2008