Series Divergent

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

Since the limit sum does not exist, the series is divergent. This is defined as a bounded or oscillating divergent series. Similarly for the value r = +1,... 

The series is also divergent but defined as an unbounded divergent series. 

The sum of two divergent series can be convergent. Similarly, the sum of a convergent series and a divergent series must be divergent. 

Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms. 

The undefined mechanism of the aldol-type Mukaiyama and Sakurai allylation reactions arose the discussion and interest in mechanistic studies [143-145]. The proposed mechanism was proved to proceed through the catalytic activation of the aldehyde and its interaction with the silyl ketene acetal or allylsilane producing the intermediate. From that point the investigation is complicated with two possible pathways that lead either to the release of TMS triflate salt and its electrophihc attack on the trityl group in the intermediate or to the intramolecular transfer of the TMS group to the aldolate position resulting in the evolution of the trityl catalyst and the formation of the product (Scheme 51). On this divergence, series of experimental and spectroscopic studies were conducted. 

J. Cizek, F. Vinette, and E. J. Weniger, Int. ]. Quantum Chem., Quantum Chem. Symp., S25, 209 (1991). Examples on the Use of Symbolic Computation in Physics and Chemistry Applications of the Inner Projection Technique and of a New Summation Method for Divergent Series. 

Ford, W. B., Studies on Divergent Series and Summability and The Asymptotic Developments of Functions Defined by MacLaurin Series, London Chelsea Publishing, 1960. 

We see that the convergence of this series is very poor and the same is true for any n. Still, by using summation methods a la Borel (see Section 3.2.2), it is possible to extract precise values of the exponents from these divergent series. In particular, Le Guillou and inn-Justin34 found the following values... 

Divergent series are not much used in physical work, while converging series are very frequently employed.1... 

The student should be able to discriminate between convergent and divergent series. I shall give tests very shortly. To simplify matters, it may be assumed that the series discussed in this work satisfy the tests of convergency. It is necessary to bear this in mind, otherwise we may be led to absurd conclusions. E. W. Hobson s On the Infinite and Infinitesimal in Mathematical Analysis, London, 1902, is an interesting pamphlet to read at this stage of our work. 

A prize was offered in France some time back for the best essay on the use of diverging series in physical mathematics. 

If y = 4, the development gives a divergent series and the theorem is then said to fail. The last four examples are logarithmic series. 

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