This book discusses some of the first principles of modern analysis. I t can be used for courses at several levels, depending upon the background and ability of the students. It was written on the premise that today's good students have unexpected enthusiasm and nerve. When hard work is put to them, they work harder and ask for more. The honors course (at the University of Wisconsin) which inspired this book was, I think, more fun than the book itself. And better. But then there is acting in teaching, and a typewriter is a poor substitute for an audience. The spontaneous, creative disorder...

At the end of the typical one quarter course on power series the students lack the means to decide 2 whether 1/(1]x ) has an expansion around any point 0, or the tangent has an expansion anywhere and the means to evaluate and predict errors. In using power series for computation the main problems are: 1) To predict a priori the number N of terms needed to do the computation with a specified accuracy; and 2) To find the coefficients aO, ., a N These are the problems addressed in the book. Typical computations envisioned are: -6 calculate with error 10 the integrals If/2 J (If/2-x)tan x dx o or...