You are here

Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis

Ovidiu Furdui
Publication Date: 
Number of Pages: 
Problem Books in Mathematics
Problem Book
[Reviewed by
Allen Stenger
, on

This is an intriguing, specialized, and scattershot book of problems in classical analysis. Each problem has a statement, a brief but revealing hint, and a complete worked solution.

The subject matter is very specialized, and deals almost exclusively with evaluation of series. The “fractional part integrals” of the title are integrals with a factor of {1/x}, where {} denotes the fractional part of a number. Because of the jumps in the integrand, most of these have an alternate formulation as a series, and switching viewpoints may allow us to evaluate the series.

The overall level of difficulty is “hard”. Although some easy problems are included, most problems are quite challenging. The author describes them as suitable for undergraduates, but I think this is optimistic. Many of them come from the Problems Section of the American Mathematical Monthly, and I worked some of them when they came out and found them quite difficult.

A serious limitation of the book is its fragmentary nature. It does not give you a complete course in anything, as many problem books do (for example, Pólya & Szegö’s Problems and Theorems in Analysis). This is a deliberate choice by the author; he says on p. xiii, “I have tried to avoid collecting too many problems that are well known or published elsewhere, in order to keep a high level of originality.”

Bottom line: an excellent book for browsing, but less suitable for a course or for reference.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.



1. Limits

2. Fractional Part Integrals

3. A Bouquet of Series

A. Elements of Classical Analysis

B. Stolz–Cesàro Lemma