Good book for convergence of series

Solution 1:

I recommend the study of Real Infinite Series by Daniel D. Bonar and Michael J. Khoury.

• The first two chapters Introduction to Infinite Series and More Sophisticated Techniques provide a firm ground of basic convergence tests and some more sophisticated results which is not that often presented in introductory material.

To get an idea what is addressed by more sophisticated we can e.g. find in section 2.4 the notation 2.15 stating:

• Define $\log_{(0)}x=x,\log_{(1)}x=\log x,\log_{(2)}x=\log(\log x)$, and, inductively, $\log_{(n+1)}x=\log(\log_{(n)}x)$.

• Theorem 2.16 (Abel-Dini Scale) For each integer $k$, the series \begin{align*} \sum_{n=N}^\infty\frac{1}{n\log_{(1)}n\log_{(2)}n\cdots\log_{(k)} n\left(\log_{(k+1)}n\right)^\alpha} \end{align*} converges if $\alpha >1$ and diverges if $\alpha \leq 1$, where $N$ is taken large enough for the iterated logarithms to be defined.

• The third chapter The Harmonic Series and Related Results presents nice information about the harmonic series and friends. E.g.

• Theorem 3.17 Let $B$ be the set of positive integers whose decimal representation do not contain the digit $9$. Then the series \begin{align*} \sum_{n\in B}\frac{1}{n} \end{align*} converges.

Here we are at a border line with the divergent harmonic series on one side and related series where we ask if they are also divergent or not.

The first three chapters provide a good basis for the study of even more sophisticated series organized in three following chapters.

• Chapter 4 Intriguing Results provides $107$ so-called gems of different nature.

Gem $29$: \begin{align*} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^n}=\int_0^1x^x\,dx \end{align*}

Gem $102$: For each real number $x$, \begin{align*} \sum_{n=1}^\infty\frac{\sin(nx)}{n} \end{align*} converges.

• Chapter 5 Series and The Putnam Competition presents challenging problems about series and their solutions.

• Chapter 6 Final Diversions gives some additional nice examples grouped in sections: Puzzles, Visuals, Fallacious Proofs and the last called Fallacies, Flaws and Flimflam.

This book is also worth reading for its interesting and informative appendices:

• Appendix A: $101$ True or False questions

This is a good possibility to check what do we really know about series.

• Example: T/F $39$ \begin{align*} \sum_{n=1}^\infty\left(1-\frac{1}{n}\right)^n \end{align*} is a divergent series.

• Example: T/F $40$ The Root Test cannot alone be used to determine conditional convergence.

Finally I would like to point at

• Appendix C: References

Here we obtain information grouped in

• Books on Infinite Series

• Books with Excellent Material on Infinite Series

• Sources for Excellent Problems Related to Infinite Series

• Pleasurable Reading

• Journal Articles

Solution 2:

Classical books are quite good when it comes to basic problems on convergence and divergence. Try Pólya's and Szegö's Problems and theorems in Mathematical Analysis which is a great treasurehouse of problems. Knopp's Theory and Applications of Infinite Series is good for applications and summability methods. William F. Osgood's Introduction to infinite series and Bromwich's Introduction to the theory of Infinite series are also great classics. A modern work of interest is Titu Andreescu and T-L Radulescu's Problems in Real Analysis which gives a great source of Problems and further references.