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Extra resources for Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications ~ Volume 106, Number 3 June, 2009 pp325-499
However, what is not well-known is that the series is conditionally convergent for z < 1 and divergent for > 1. Although this issue has been discussed in Refs.  and , since it is critical for understanding the concepts presented in this work, we present its proof in the following lemma. k Lemma The geometric series ∞ k=0 z is absolutely convergent for |z| < 1, conditionally convergent for z < 1 and |z| > 1 and divergent for z > 1. For the latter case, the series must be regularized by removing the infinity in the remainder in order to obtain a finite value.
Math. Phys. 38, 1347 (1997). hep-th/9405067 9. : Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108 (1995) 10. : Minimal length uncertainty relation and ultraviolet regularization. Phys. Rev. D 55, 7909 (1997) 11. : Minimal length uncertainty relation and the hydrogen atom. J. Phys. A 32, 7691 (1999) 12. : Minimal length uncertainty principle and the transPlanckian problem of black hole physics. Phys. Rev. D 59, 044005 (1999) 13. : Wave-particle duality at the Planck scale: freezing of neutrino oscillations.
Previously, we assigned a value of −1/(i + 1)! for each element i when we applied the partition method in developing a general formula for the ak . If we compare the series in (14) with that in (30), then we see that the coefficient of t i is −ai instead of −1/(i + 1)!. Furthermore, when we develop a power series in t , we need to take into account the additional factor of t outside the series in (30). That is, the coefficient of t i obtained when we apply the partition method to the series in (30) will represent the coefficient of t i+1 overall.
Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications ~ Volume 106, Number 3 June, 2009 pp325-499 by Springer Netherlands