Almost Harmonic

Calculus Level 3

The harmonic series is the sum of the reciprocals of all natural numbers; mathematically, \(\displaystyle\sum_{i=1}^\infty\dfrac{1}{i}.\) This series has an infinite sum, as first proved by Nicole Oresme in the 14th century.

Find \(\lfloor1000N\rfloor,\) where \(N\) is the largest possible non-negative \(K\) for which the following sum does \(\textbf{not}\) diverge to \(-\infty.\) \[\sum_{i=1}^\infty\dfrac{1}{i}-K\]

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