The Limit Of The Exponent

Calculus Level 3

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

Determine limnan \displaystyle\lim_{n \rightarrow \infty} a_n .

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