x2令?（x)?x?sinx?,x?(0,1).则?（'x)??x?1?cosx2令g(x)??'(x)??x?cosx?1,0?x?1?g'(x)??1?sinx?0?g(x)在（0，1）上单调递减，即?‘（x)在（0,1）上单调递减又?0?x?1??'(x)??'(0)?0??(x)在（0,1)上单调递减又?0?x?1??(x)??(0)?0恒成立又?0?an?1a??(an)?0,即an?sinan?n22an?an?1?221再证明a1?时，an?n222anaa由an?1??n?1?n2an2又?an?an?1?an?2???a2∵
?02?a1naaaaaaaaaaa1当n?2时，an?a1?2?3?n?a1?1?23??a1?1?11?1?na1a2an?122222222?1?2?1???2?1??n?1??2?22n?12n1?an?n2??????????14分
n?1证法二：利用数学归纳法和当x?(0,)时，y?sinx?x单调递增?ak?1?f(ak)?621ak?sinak?ak2