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Today's calculation of Integral 374

Source: 1996 Osaka University entrance exam/Science

September 7, 2008
calculusintegrationlogarithmslimitinequalitiescalculus computations

Problem Statement

Let n≥2 n\geq 2 be positive integers. (1) Prove that n\ln n \minus{} n\plus{}1<\sum_{k\equal{}1}^n \ln k<(n\plus{}1)\ln n \minus{} n\plus{}1. (2) Find lim⁡n→∞(n!)1nln⁡n \lim_{n\to\infty} (n!)^{\frac{1}{n\ln n}}.
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