用无穷矩阵方程证明雅各布·伯努利的幂和公式  

Proving Jacob Bernoulli's Power Sum Formula with Infinite Matrix Equation

作　　者：杨庆玺[1] 唐军强[1] YANG Qingxi;TANG Junqiang(Basic Teaching Department,Jiaozuo University,Jiaozuo,Henan 454000,China)

机构地区：[1]焦作大学基础教学部,河南焦作454000

出　　处：《焦作大学学报》2023年第4期63-65,共3页Journal of Jiaozuo University

摘　　要：将自然数的各阶幂和所满足的一个递推关系式写成矩阵方程的形式,通过对该矩阵求逆,发现逆矩阵中的元素可以用Bernoulli数很好地描述,从而获得各阶幂和所满足的一种用Bernoulli数表示的通用公式。通过将公式中的组合数用二项式定理展开,比较n的幂次的系数,证明了与雅各布·伯努利的公式的一致性。A recursive relationship satisfied by the sum of powers of natural numbers is written in the form of a matrix equation.By inverting the matrix,it is found that the elements in the inverse matrix can be well described by Bernoulli numbers,thus obtaining a universal formula expressed by Bernoulli numbers for the sum of powers of each order.By expanding the number of combinations in the formula using the binomial theorem and comparing the coefficients to the power of n,the consistency with Jacob Bernoulli's formula was proved.

关 键 词：二项式定理 无穷矩阵方程 自然数方幂和 伯努利数 伯努利多项式 

分 类 号：O156[理学—数学]