关于 和级数型展开式的规律分析xxxxxx 大学数学与计算机科学学院数学与应用数学专业 2025 级数学班指导老师:xx【摘要】: 众所周知,的幂级数展开形式为: 其中,取时,得到: 下面,我们问:级数 , , , , 是什么?事实上,讨论幂函数级数更为方便,因为我们可以用微积分作工具。对 令 ,由于右端幂级数的收敛半径为:,故对有定义。特别地,当时,有 。本文通过计算论证,得出: ,其中是关于的一个次多项式这一结论。将多项式的系数作成一个无穷阶的矩阵: ,文中对矩阵的各行、列、斜方向进行分析、总结,从而得出结论。【关键词】:幂级数的展开;微积分;递推公式;多项式;矩阵About and series type expansion analysis of the lawxxxGrade 2025, Math class, mathematics and applied mathematics major,School of Mathematics & Computer science of xxxxxxxInstructor:xxAbstract: As everyone knows , form of power series expansion :Among them, take , are:Below, we ask: what is the progression , , , , .In fact, research of the power function progression more convenient, because we can use calculus as a tool.For , , due to the radius of convergence of the right end power series are as follows: , Therefore, to is defined.In particular, when , .Demonstrated by calculation, this paper concludes that: , Among them is about x k polynomial of the conclusion.Will of polynomial coefficient into a matrix of infinite order, remember to , of the matrix rows, columns and oblique direction of analysis, summarized, to draw conclusions.Key words: The development of power series ; Differential and integral calculus ;The recursive formula ; Polynomial ; Matrix 在微积分中,我们知道:更一般地,有:下面,我们首先来讨论,当时, :令 猜想 1: 其中 。下面,我们需要验证猜想 1 是否正确。答案:猜想 1 正确。在验证猜想 1 之前,我们先来讨论当时的情形,即当时,有:令 由微积分知: 有上述运算可得下面定理:定理 1 :满足下列递推公式: 证明: , 。 由定理 1...