There Are Cancellations?

Algebra Level 3

\[\large a_n=\sqrt{ 1+\left(1+\dfrac{1}{n} \right)^2 }+\sqrt{ 1+\left(1-\dfrac{1}{n} \right)^2 }\]

Consider a sequence \(a_n\), \(n\geq1\) as defined above. Compute \(\dfrac{1}{a_1}+\dfrac{1}{a_2}+\cdots+\dfrac{1}{a_{20}}\).