Integer Part of the Partial Sums for the Divergent p-Series in Closed Form

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 (Formula presented.), 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.