We prove a combinatorial identity containing k-order differences of sequences, as well as binomial coefficients. The Euler summation operation implies the calculation of all order differences of terms of the initial series. The regularity of the summing function means the coincidence with the “ordinary” sum of the series, provided that this sum exists. The proved combinatorial identity allows one to easily prove the regularity of the Euler summation operation.

中文翻译：

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数值级数的欧拉求和

我们证明了包含序列的k阶差异以及二项式系数的组合恒等式。欧拉求和运算意味着计算初始序列项的所有阶差。求和函数的规则性是指与该序列的“普通”和的巧合，前提是该和存在。证明的组合身份使人们可以轻松证明欧拉求和运算的规律性。