Abstract Although no known closed-form expression exists for the Nth partial sum of the p-series, we show that a closed-form expression can at least be found for the integer part of these partial sums, where , for specific sequences of integers N that are dependent on the value of p. The closed-form expressions follow from an application of the upper and lower bounds obtained from the integral test for the partial sums of series with positive terms.

中文翻译：

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封闭形式的发散 p 系列的部分和的整数部分

摘要 尽管 p 系列的第 N 个部分和不存在已知的封闭形式表达式，但我们表明至少可以找到这些部分和的整数部分的封闭形式表达式，其中 ，对于特定的整数序列 N取决于 p 的值。封闭形式的表达式来自于应用从具有正项的序列的部分和的积分测试获得的上下界。