Fixing singularities with fun(ctions)

Calculus Level 5

\[f\left( x \right) =\tan { \left( x \right) } \tan { \left( x+\alpha \right) }\]

Let \({ a }_{ n }\) denote the \(n^\text{th}\) smallest positive value of \(\alpha\) that will make \(f\left( x \right)\) continuous over all real values of \(x\). Evaluate the following sum:

\[\sum _{ n=1 }^{ \infty }{ \dfrac { { \left( \tan { ( n ) } \tan { ( n+{ a }_{ n } ) } \right) }^{ n } }{ { a }_{ n } } }.\]

Round your answer to the nearest three decimal places.

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