Today's calculation of Integral 228
Source: Tokyo Medical and Dental University entrance exam 1972
October 4, 2007
calculusintegrationtrigonometrylimitcalculus computations
Problem Statement
Let x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots).
(1) Show that x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}.
(2) Find the value of nx_nx_{n \minus{} 1}.
(3) Show that a sequence is monotone decreasing.
(4) Find .