Coefficient of expansion and limits

Let \({ A }_{ n }\) be the coefficient of \( {x}^{-n+3} \) in the expansion of \( \left(x+ \frac {1}{x} \right)^{n+1} \). Determine the value of \( \displaystyle{\sum _{ n=1 }^{ \infty }}{ \frac { 1 }{ { A }_{ n } } } , \) where \(n\) is a positive integer.

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