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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