摘要
前 n 个自然数的 m(m∈N)次幂的和的计算问题,是世界著名100个初等数学问题之一。本文得出一个新的计算公式,并且用差分方法。给出了严格证明。公式为:1~m+2~m+3~m+…+n^m=(n+1)sum from (k=1) to m sum from (r=0) to (m-1)((-1)~rC_k^r(k-r)~m)/((k+1)1)n^(k)(k=1,2,…,m),m 为任意自然数(m,n∈N)。
It is one of one hundred elementary mathematicl problems of the world to compute the sum of first n natural numbers to power m(m,n ∈ N).This paper shows a new computing formula which was obtained with the difference method and strictly proved.The formula is follows: 1~m+2~m+3~m+…+n^m=(n+1)(m ∈N,N∈N)
关键词
自然数
m次幂的和
求和公式
Natural number
Sum formula
Difference method
