# Fixing singularities with fun(ctions)

Calculus Level 4

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

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:

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