The Recursion Is Trapped In The Exponent!

Calculus Level 2

The sequence $$\{ {a}_{n}\}$$ satisfies $$\left(\frac{2n+1}{2n-1}\right)^{a_n} =e^2.$$ Find $$\lim_{n \rightarrow \infty}\frac{2a_n}{n+1}$$.

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