马德瓦的π级数在现代语境中的应用
Madhava's Pi-Series in the Modern Context
AI总结：
本文通过现代符号和定义重新诠释马德瓦的π级数证明，强调其简洁性和基于弧长的原始性，对比传统基于莱布尼茨的证明，突出马德瓦方法的先进性。
摘要：
π级数归因于马德瓦、格雷戈里和莱布尼茨，这是基于其发现时间顺序的。尽管这承认了马德瓦，一位14世纪生活在印度喀拉拉邦尼拉河沿岸的数学家，是第一个发现该级数的人，但现代对π级数的证明是基于著名德国数学家莱布尼茨在17世纪给出的证明框架，使用了许多在莱布尼茨去世后才发展出的微积分概念和严谨定义。我们同样将这种好处延伸到马德瓦的思想，呈现马德瓦在《优陀耶德瓦》中给出的π级数证明，即upapatti，即证明，使用现代符号和定义。我们详细阐述马德瓦的证明在三个地方：（a）证明弧段和与弧段近似和相同；（b）证明收敛性；（c）通过迭代简洁地证明幂和近似。马德瓦的方法通过适当修改可以用于推导正弦和余弦级数。马德瓦还提供了修正项以加速π级数的收敛性。我们比较了两种证明，即马德瓦的证明和基于莱布尼茨思想的流行π级数证明，发现它们不同，因为马德瓦的基于弧长，而莱布尼茨的基于弧线下方的面积，在两个证明中使用的高级微积分概念数量方面，马德瓦的证明因其简洁性而脱颖而出，从而为它成为π级数的原始证明提供了依据。
英文摘要：
The $π$-series is attributed to Madhava, Gregory and Leibniz based on the chronology of its discovery. While this acknowledges the fact that Madhava of Sangamagrama an Indian mathematician who lived in Kerala in the 14th century along the banks of river Nila, was the first to discover the series, the modern day proof for the $π$-series is based on the outline of the proof given by the famous German mathematician Gottfried Wilhelm Leibniz in the 17th century, using concepts and rigorous definitions of Calculus many of which were developed after his lifetime. We extend the same benefit to Madhava's ideas and present Madhava's upapatti i.e proof for the $π$-series as given in Jyesthadeva's Yuktibhasha using modern notation and definitions. We elaborate Madhava's proof at three places (a) to prove that the arc-bit sum is same as the sum of arc-bit approximations (b) to prove convergence and (c) to concisely prove by iteration the power sum approximation. Madhava's method with appropriate changes can be applied to derive the sine and cosine series as well. Madhava also provides the correction terms to accelerate the convergence of the $π$-series. We compare the two proofs i.e Madhava's and the popular proof for $π$ which is based on Leibniz's ideas and observe that they are different since Madhava's is based on the length of the arc where as Leibniz's relies on the area under the arc and in terms of the number of concepts from advanced calculus used in the the two proofs Madhava's proof stands out for its simplicity, thus making a case for it to be appreciated, popularised and taught as the original proof of the $π$-series.
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