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Today's calculation of Integral 808

Source: 2012 Kyusyu University entrance exam/Science, 2nd exam

March 14, 2012
calculusintegrationlogarithmslimitcalculus computations

Problem Statement

For a constant cc, a sequence ana_n is defined by an=∫c1nxn−1(ln⁡(1x))ndx (n=1, 2, 3, ⋯ ).a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).
Find lim⁡n→∞an\lim_{n\to\infty} a_n.
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