| Popis: |
Yes, but not quite: Heinrich Bruns (1884) (our Bruns!) made the race, but Poincare (1890, 1892–1899) went much farther and deeper. He proved that “most” series used in celestial mechanics were divergent but nevertheless perfectly useful. In fact, he recognized that mathematical convergence or divergence may be quite irrelevant for numerical convergence: the series $$ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} \ldots $$ is convergent but numerically practically useless because the convergence is so slow. On the other hand, the series $$ frac{{{e^x}}}{x}\left( {1 + \frac{{1!}}{x} + \frac{{2!}}{{{x^2}}} + \ldots } \right) $$ is divergent but numerically superbly useful as we shall see in sec. 2 of this paper. |
