Can anybody please suggest me a good book for convergence of series where I could find good questions on sum of series, sum of alternating series etc. I already have solved standard books like Tom apostol's calculus. Knopp's theory etc. Basically I'm looking for a book having more questions and less chit-chat. I want to solve as much problems as I can.
[Math] Good book for convergence of series
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Best Answer
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.
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.
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.
This book is also worth reading for its interesting and informative appendices:
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
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