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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

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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. [4] and [5], 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.

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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.

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