There Are Cancellations?

Algebra Level 3

an=1+(1+1n)2+1+(11n)2\large a_n=\sqrt{ 1+\left(1+\dfrac{1}{n} \right)^2 }+\sqrt{ 1+\left(1-\dfrac{1}{n} \right)^2 }

Consider a sequence ana_n, n1n\geq1 as defined above. Compute 1a1+1a2++1a20\dfrac{1}{a_1}+\dfrac{1}{a_2}+\cdots+\dfrac{1}{a_{20}}.

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