Power Series and the Exponential Function

Abstract

As with integrals, power series (though fascinating) are a tool here and are not pursued extensively. (However, some special kinds of power series, those with many zero coefficients or with integer coefficients, are examined in some detail in Chapters XVI and XVII.) For in-depth treatises on power series the reader should consult Knopp [1951] or Bromwich [1926]. To make our definitions and get started only one simple result is needed: $${{\lim }_{{n \to \infty }}}{{n}^{{1/n}}} = 1 \leqslant and \leqslant for \leqslant any \leqslant a > 0,{{\lim }_{{n \to \infty }}}{{n}^{{1/n}}} = 1.$$ .