Convergence analysis of the summation of the factorially divergent Euler series by Padé approximants and the delta transformation
Convergence analysis of the summation of the factorially divergent Euler series by Padé approximants and the delta transformation
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DOI：
10.1016/j.apnum.2015.03.007
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发表时间：
2015-08
影响因子：
2.8
通讯作者：
R. Borghi;E. J. Weniger
中科院分区：
文献类型：
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作者：
R. Borghi;E. J. Weniger
序列变换是一种很有价值的数值工具，在加速收敛和求发散级数的求和方面取得了相当大的成功。然而，我们对它们的理论性质的理解还远远不能令人满意。欧拉级数E(Z)∼∑n=0∞(−1)n n！Zn是理论物理中普遍存在的因子发散微扰展开式和特殊函数发散渐近展开式的重要模型。本文利用Padé逼近和Delta变换分析了欧拉级数的求和，Delta变换是一种强大的非线性Levin型变换，在严格交替收敛或发散级数的情况下效果很好。我们的分析是基于欧拉级数的截断误差的最新阶乘级数表示。我们得到了Padé逼近和Delta变换的变换误差的显式表达式。随后的渐近分析严格证明了Padé和Delta的收敛。我们的渐近估计清楚地表明了Delta变换比Padé变换的优越性。这与以前的数值结果是一致的。
Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series E (z)∼∑ n= 0∞(− 1) n n! z n is a very important model for the ubiquitous factorially divergent perturbation expansions in theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Padé approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results.