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Today's calculation of Integral 831

Source: 1977 Tsukuba Univrersity entrance exam

July 3, 2012
calculusintegrationlimitinductioncalculus computations

Problem Statement

Let nn be a positive integer. Answer the following questions.
(1) Find the maximum value of fn(x)=xne−xf_n(x)=x^{n}e^{-x} for x≥0x\geq 0.
(2) Show that lim⁡x→∞fn(x)=0\lim_{x\to\infty} f_n(x)=0.
(3) Let In=∫0xfn(t) dtI_n=\int_0^x f_n(t)\ dt. Find lim⁡x→∞In(x)\lim_{x\to\infty} I_n(x).
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