This book is based on several courses taught by the author at the University of Michigan between 1908 and 1912. It covers two main topics: asymptotic series and the theory of summability. The discussion of nowhere convergent asymptotic series includes the so-called MacLaurent summation formula, determining asymptotic expansions of various classes of functions, and the study of asymptotic solutions of linear ordinary differential equations. On the second topic, the author discusses various approaches to the summability of divergent series and considers in detail applications to Fourier series.
Studies on Divergent Series and Summability: The MacLaurin sum-formula, with introduction to the study of asymptotic series The determination of the asymptotic developments of a given function The asymptotic solutions of linear differential equations Elementary studies on the summability of series The summability and convergence of Fourier series and allied developments Appendix Bibliography The Asymptotic Developments of Functions Defined by MacLaurin Series: Preliminary considerations. First general theorem The theorem of Barnes MacLaurin series whose general coefficient is algebraic in character Second general theorem Auxiliary theorems MacLaurin series whose general coefficient involves the reciprocal of a single gamma function; Functions of exponential type MacLaurin series whose general coefficient involves the reciprocal of the product of two gamma functions; Functions of Bessel type Determination of the asymptotic behavior of the solutions of differential equations of the Fuchsian type Bibliography.