An algebra problem by Mateus Gomes

Algebra Level 3

\[ \dfrac{1}{\log_2(2010!)}+\dfrac{1}{\log_3(2010!)}+\cdots+\dfrac{1}{\log_{2009}(2010!)}+\dfrac{1}{\log_{2010}(2010!)} = \, ? \]

\[\]Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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