Weird function

Algebra Level 4

Consider the function f(n)=4n+4n212n+1+2n1f(n)=\dfrac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}} .

Evaluate k=140f(k)\displaystyle\sum_{k=1}^{40} f(k).

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