The Limit Of The Exponent

Calculus Level 3

It is well known that the sequence \( ( 1 + \frac{1}{n} ) ^ n \) approaches \(e \) from below, and the sequence \( ( 1 + \frac{1}{n} ) ^ {n+1} \) approaches \( e \) from above. Hence, for each \(n\), there is a unique value \( a_n \) between 0 and 1 such that \( ( 1 + \frac{1}{n}) ^ {n+ a_n} = e \).

Determine \( \displaystyle\lim_{n \rightarrow \infty} a_n \).

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