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About this book

This book contains a multitude of challenging problems and solutions that are not commonly found in classical textbooks. One goal of the book is to present these fascinating mathematical problems in a new and engaging way and illustrate the connections between integrals, sums, and series, many of which involve zeta functions, harmonic series, polylogarithms, and various other special functions and constants. Throughout the book, the reader will find both classical and new problems, with numerous original problems and solutions coming from the personal research of the author. Where classical problems are concerned, such as those given in Olympiads or proposed by famous mathematicians like Ramanujan, the author has come up with new, surprising or unconventional ways of obtaining the desired results. The book begins with a lively foreword by renowned author Paul Nahin and is accessible to those with a good knowledge of calculus from undergraduate students to researchers, and will appeal to all mathematical puzzlers who love a good integral or series.

Table of Contents


Chapter 1. Integrals

A challenging question: How would we solve the Basel problem using the integral result?
Cornel Ioan Vălean

Chapter 2. Hints

The four integrals are related to each other, and for deriving the squared log version we need the simple log version, then for deriving the cubic log version, we need the squared log version and so on. For an elementary approach, use wisely the integration by parts.
Cornel Ioan Vălean

Chapter 4. Sums and Series

In their hearts humans plans their course, but the LORD establishes their steps.”—Proverbs 16:9
Cornel Ioan Vălean

Chapter 5. Hints

For a first solution to the point i) of the problem, make use of the result in (1.​1), and for a second solution, wisely employ the Beta function.
Cornel Ioan Vălean


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