In this post, we will see the book *Infinite Series Ramifications* by G. M. Fichtenholz. This book is the Course 4 of the *Pocket Mathematical Library* series.

# About the book

The present volume of The Pocket Mathematical Library continues the study of infinite series begun in its companion volume Infinite Series: Rudiments, by the same author. Together the two volumes give a detailed treatment of the theory of numerical series, i.e., infinite series whose terms are numbers. The picture is then completed by a third volume, entitled Functional Series, which, as its name implies, is devoted to the study of infinite series whose terms are functions. The set of three volumes makes up a comprehensive treatise on all aspects of a key topic of pure and applied mathematics.

As in the companion volume, the problems appearing at the end of each section constitute an important part of the course, and should not be neglected by the serious student.

The book was translated from Russian by Richard Silverman and was published in 1970.

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You can get the book here.

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# Contents

## Chapter 1. Operations on Series 1

l. Associativity of Convergent Series 1

2. Commutativity of Absolutely Convergent Series 5

3. Riemann’s Theorem 7

4. Multiplication of Series 13

5. Toeplitz’s Theorem 21

6. The Theorems of Mertens and Abel 26

## Chapter 2. Iterated and Double Series 30

7. Iterated Series 30

8. Double Series 35

9. Examples 43

10. Power Series in Two Variables 54

## Chapter 3. Computations Involving Series 61

11. General Remarks 61

12. Examples 63

13. Euler’s Transformation 70

14. The Transformations of Kummer and Markov 78

## Chapter 4. Summation of Divergent Series 87

15. Introduction 87

16. The Method of Power Series 90

17. The Method of Arithmetic Means 98

18. Application of Generalized Summation to Multiplication of Series 110

19. Other Methods of Generalized Summation 113

20. The Methods of Borel and Euler 121

Index 129

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