Fixing singularities with fun(ctions)

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

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.