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Today's calculation of Integral 262

Source: 1977 Tsukuba university entrance exam

January 10, 2008
calculusintegrationlimitfunctionderivativecalculus computations

Problem Statement

Answer the following questions for positive integer n. n. (1) Find the maximum value of f_n(x) \equal{} x^ne^{ \minus{} x} for x≥0. x\geq 0. (2) Show that \lim_{x\to\infty} f_n(x) \equal{} 0. (3) Let I_n (x) \equal{} \int_0^x f_n(t)\ dt, find lim⁡x→∞In(x). \lim_{x\to\infty} I_n(x).
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