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$I_{n} = \int_{0}^{1} x^{n}e^{x-1}dx$

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0 < I_{n} < \frac{1}{n+1}

The lower bound is obvious but my attempts to get an upper bound have been unsuccessful.

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When 0 < x <1 note that x^{n}e^{x-1} < x^{n} and if f < g then \int f < \int g