Logs on fire!

Calculus Level 5

\[S=\log _{ 2 }{ e } -\log _{ 3 }{ e } +\log _{ 4 }{ e } -\log _{ 5 }{ e } +\log _{ 6 }{ e } \dots \dots \dots \]

Find \(\left\lfloor 1000S \right\rfloor \)

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