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Today's calculation of Integral 871

Source: 2013 Tohoku University entrance exam/Medicine etc.

March 15, 2013
calculusintegrationtrigonometrylimitlogarithmscalculus computationsDefinite integral

Problem Statement

Define sequences {an}, {bn}\{a_n\},\ \{b_n\} by an=∫−π6π6ensin⁡θdθ, bn=∫−π6π6ensin⁡θcos⁡θdθ (n=1, 2, 3, ⋯ ).a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).
(1) Find bnb_n.
(2) Prove that for each nn, bn≤an≤23bn.b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.
(3) Find lim⁡n→∞1nln⁡(nan).\lim_{n\to\infty} \frac 1{n}\ln (na_n).
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