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.
Credits to original uploader.
You can get the book here.
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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