# 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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