AN INTRODUCTION TO THE THEORY OF INFINITE SERIES

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This historic book may contain numerous typos and missing text. Purchasers can usually download a free scanned copy of the original work from the publisher. The 1908 edition is not indexed or illustrated. In numerical work, only asymptotic series are used; these series have terms that initially decrease to a minimum before increasing again. When summing to a point where the terms are sufficiently small, an approximation can be made with a degree of accuracy represented by the last term retained. Many series suitable for numerical calculations exhibit this property. A significant class of these series is employed by astronomers to calculate planetary positions. Poincare proved that these series do not converge, yet the calculations align with observational results, which can be explained by his theory of asymptotic series. However, mathematicians have also used series where terms do not decrease and may increase indefinitely. Euler, for instance, considered the "sum" of a non-convergent series as the finite numerical value derived from the series' expansion. He defined the "sums" of various series, illustrating how inverting limits can yield a definite value, depending on the order of operations.