1 D9ADA6 2002 BOCZA1DCABD9ADB8A3AIAOAKCBA4B5A6 A3D3D8BLADB1C7A5DFA72002CK10AOA5 1A0ARB1AIAPB6AXCGD0A0A7CVBYAVDIAGABA2 2A0D2β =supE,β /∈ E,D6AXC4ARAZACCTD8CC{x n }A7CNBMA8DJβ A9ALCYβ ∈ E A7AQCPAACXBJAA 3A0BZC8A8A31A5AKD0CVBYA1A7CVBYAVD8CCA9A32A5BHAKD0CVBYAUAJBHAKA7CVBYAVD8CCA9A33A5 BQBHAKD0CVBYALBHAKA7CVBYAVD8CCA9A34A5BQAJBHAKD0CVBYALAJBHAKA7CVBYAVD8CCA7CNAZD0A0A7CVBY B2AKA8A2 4A0D6AXA8D7C9D8CCAFAKD0CVBYBIA7CVBYA9CRAM +∞AVD8CCAFAKA7CVBYA7CRAM ?∞AVD8CCAFAK D0CVBYA2 5A0CQD8CCAVD0A0A7CVBYA8 (1) x n =1? 1 n ;(2)x n = ?n[2 + (?2) n ]; (3) x 2k = k,x 2k+1 =1+ 1 k ,k=1,2,3, ...;(4 n radicalbigg |cos nπ 3 |. 6A0AXCGA8ATB0AKBYD8CCAFAKBMA8A2 7 A0D6B7A2CSBTDDB1C7AVDGBVA8CYBW?CSBTCCBAANC2CSBTCCA7BXBFARAFAACYBWDGBV [a 1 ,b 1 ] ? [a 2 ,b 2 ] ? [a 3 ,b 3 ] ? ... CUAZBKBWDGBVb n ? a n → 0(n →∞)CUAZA7BXBFARAFAAD6BZC8DACGA2 8 A0CY{x n }A1BYA7COAJAH∞DJBMA8A7AQAFASAPCABDB0CC x g(n) →∞,x h(n) → a(a DJCHAKA8D8), CNAZ{g(n)}BI{h(n)}D5CABD?BCATB0AVAWAVD8CCA2 9A0AKBYD8CC{x n }CYAJD7C9A7AQAFASAPCABDB0CC x g(n) → b,x h(n) → a(n →∞), CNAZ a negationslash= b. 10A0CYAPCSBT[a,b]CIAVCABDD8CC{x n }BN{y n }CEB2 lim n→∞ (x n ? y n )=0A7AQAPAQCAD8CCAZCJAT APC1AKA9DHB2AG g(n)AVB0CCA7D4{x g(n) }BN{y g(n) }D7C9AMDHAGBMA8A2 11A0AXCGA8CYa n > 0(n =1,2,3, ...)CO lim n→∞ a n · lim n→∞ 1 a n =1 A7AQD8CC{a n }D7C9A2 12A0AXCGA8CY{a n }DJATB0D8CCA7AQ lim n→∞ a n = lim n→∞ a n . *13A0D6AXCGA8CVBYANC7A0ATB0AKBYB1C7A0?CSBTDDB1C7A0AKA8B9BBB1C7(BGCL -AHC6B4A7Heine- Borel)A0C0AYB1C7A0AYCFABB1C7(WeierstrassA7A0B4DBDEC5DB)A0C3A4A3Cauchy)D7C9B1C7CACAAWBSA2 14A0BP a n = braceleftBig (1 + 1 n ) n bracerightBig ,b n = braceleftBig (1 + 1 n ) n+1 bracerightBig ,n∈ N + . AXCGA8(1)D8CC{a n }ATB0ASBRAKBYA9(2) D8CC{b n }ATB0BUD1AKBYA9A33A5 lim n→∞ a n = lim n→∞ b n = e. CNAZ e =2.71828···D5B1CWB3D8AVAXA2 15A0AJD0DFBXCDAXCGA7CCD8CCD7C9COAKBEDHBMA8A3AUBMA8ACAMDJCMC5ALD8γ =0.577215664901 53286060651···)A8 (1) D n =1+ 1 2 + 1 3 +···+ 1 n ?ln n;(2)E n =1+ 1 2 + 1 3 +···+ 1 n ?ln(n +1). 16A0D2 x 1 =sina, x n+1 =sinx n ,n=1,2, .... AXCGA8D8CC{x n }D7C9A2