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Monday, January 3, 2011

Infinite series ( and Euler's identity revisited )

The importance of infinite series and sequences can not be underestimated in my opinion, they literally pop up everywhere. To my surprise however, there isn't a single course at the Open University ( or any other university to my knowledge ) that deals exclusively with the topic of 'Infinite Series'. Instead the theory is stuffed away somewhere else, as if it is not really important. Like in M208 for example. - I have been searching for books on the subject and there -is- a ( recent ) book on the subject. It is called 'Real Infinite Series' by D. Bonar / M. Khoury published my TMAA. Tests not in M208 but in this book are for example Raabe's Test, Rummer's Test. Cauchy's Condensation Test, Abel's Test, and Dirichlet's Test as well as Bertrand's Test. It includes an entire chapter on the harmonic series with different divergent proofs. In the appendix there is an overview of the literature on infinite series.

This video shows the proof of $e^{i\pi}+1=0$ using infinite series.

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