1 CKADDD 2002 D0BYA1COAGCKADC3DAATB6AVAPA6ANA8 A5CJAQCXDGDBCTA9 2002BA10B6A7 1A0DEAQCMBXB9C4BTD3 aCECJAQ{a n }BJCXDGBPARAFD3C8AZAF A51A7BRAZD8BKBTCAε>0A9?N ∈N + ,BG n>NC7A9AUBKCB|a n ? a| <εB1AIAD A52A7?ε>0A9?N ∈N + ,BG n>NC7A9AXD8BKBTCAa n A9C9AUBKCB|a n ? a| <εB1AIAD A53A7BRAZCBBPBJ10 ?10 A9AUBKCB|a n ? a| < 10 ?10 CPB1AIA2 2A0CMB5DEAQAQCIBQABAPBTD3 a CECJAQ{a n }BJCXDGBPARA2 A51A7?ε>0A9?N ∈N + ,BG n ≥ N C7A9AUBKCB|a n ? a| <εB1AIAD A52A7?ε>0A9?N ∈N + ,BG n>NC7A9AUBKCB|a n ? a|≤εB1AIAD A53A7?ε>0A9?N ∈N + ,BG n>NC7A9AUBKCB|a n ? a| <kεB1AIA9BDBJ k CEAMCABDAYCJAD A54A7?ε>0A9?n ∈N + ,C9AUBKCB|a n+p ? a| <εBRBTAQBJ p ∈ N + BQB1AIAD A55A7C4BDCJAQ{ε k } ∞ k=1 AP0D3CXDGA9?ε k > 0A9?n k ∈N + ,BGn ≥ n k C7A9AUBKCB|a n ?a| <ε k B1AIAA 3A0BW{a n }B0{b n }CEALCABWBZCJAQA9CQB1BJCLB0CWCEC4BWBZAFD3C8AZAFBWBDBJAMCABWBZA9CQB1 BJCLB0CWAYBVCMAF 4A0AUACBPB1BFAACIAC A51A7CJAQ{x n }AUAP aD3CXDGAD A52A7CJAQ{x n }AUCHAJAD A53A7CJAQ{x n }D8DIA2 5A0AU ε ? N B1AGAHBEDEAQCXDGD3ASA2 (1) lim n→∞ 1 n 2 + n ; (2) lim n→∞ 1 n 4 ? n ; (3) lim n→∞ ( √ n +1? √ n); (4) lim n→∞ 10 n n! ; (5) lim n→∞ n! n n ; (6) lim n→∞ 1 n sin nπ 2 . 6A0DEAQDHAWCEC4BDBRAFBWBDBRA9BJCBB8BEB5ADBWAUBDBRA9BJA5B8BZAHA2 A51A7BW lim n→∞ a n = A A9B9 lim n→∞ |a n | = |A|; A52A7BW lim n→∞ |a n | = |A|A9B9 lim n→∞ a n = A; A53A7BW lim n→∞ |a n | =0A9B9 lim n→∞ a n =0; A54A7BW lim n→∞ a n = A A9B9 lim n→∞ a n+1 = A; A55A7BW lim n→∞ a n = A A9B9 lim n→∞ a n+1 a n =1; A56A7BW?α>0, lim n→∞ αa n = αA A9B9 lim n→∞ a n = A. 7A0CGBEB5AC 2 A51A7BW x n > 0, lim n→∞ a n = aA9B9 lim n→∞ √ a n = √ a. A52A7BW lim n→∞ a n = a A9B9 lim n→∞ 3 √ a n = 3 √ a. A53A7 lim n→∞ a n = aBJB5C2AMAKCUDCCE lim k→∞ a 2k = aA9 lim k→∞ a 2k+1 = a. A54A7BW x n > 0, lim n→∞ n √ a n = q<1A9B9 lim n→∞ a n =0. A55A7BW x n > 0, lim n→∞ n √ a n = l>1A9B9 lim n→∞ 1 a n =0. 8A0BLDEAQCJAQBJCXDGAC (1) lim n→∞ (n +1)(n +2)(n +3) 5n 3 ; (2) lim n→∞ 3 n +(?2) n 3 n+1 +(?2) n+1 ; (3) lim n→∞ parenleftbig 1+2+... + n n +2 ? n 2 parenrightbig ; (4) lim n→∞ √ n( √ n +4? √ n); (5) lim n→∞ √ 2 4 √ 2 8 √ 2... 2 n √ 2; (6) lim n→∞ n radicalbig 2+sin 2 n; (7) lim n→∞ parenleftBig 1 n 3 +1 + 4 n 3 +2 + ... + n 2 n 3 + n parenrightBig ; (8) lim n→∞ parenleftBig 1+ 1 n +1 parenrightBig n ; (9) lim n→∞ (1 ? 1 n ) n ; (10) lim n→∞ (1 + 1 n ? 4 ) n+4 ; (11) lim n→∞ n radicalbig 2sin 2 n +cos 2 n; (12) lim n→∞ ( n +1 n +3 ) 3n ; (13) lim n→∞ n summationdisplay k=1 1 k(k +1)...(k + m) (m =1,2, ...); (14) lim n→∞ ( n 3 ? 1 n 3 ? 2 ) 4n 3 ; (15) x n =(1? 1 2 2 )(1 ? 1 3 2 )...(1 ? 1 n 2 ); (16) lim n→∞ n summationdisplay k=1 (?1) k?1 3 k?1 ; (17) lim n→∞ 2n+1 summationdisplay k=1 1 √ n 2 + k ; (18) lim n→∞ parenleftBig 1+ 1 2 + ... + 1 n parenrightBig 1 n ; (19) lim n→∞ parenleftBig 1 1 · 2 ·3 + 1 2 · 3 ·4 + ... + 1 n(n +1)(n +2) parenrightBig ; (20) lim n→∞ 1 ·3 · 5···(2n ? 1) 2 · 4 ·6···(2n) ; (21) lim n→∞ parenleftBig 1 ? 1 1+2 parenrightBigparenleftBig 1 ? 1 1+2+3 parenrightBig ... parenleftBig 1 ? 1 1+2+3+... + n parenrightBig . 9A0BBARDEAQCJAQBJCHAJA6AC (1)x n = 1 3+1 + 1 3 2 +1 + ... + 1 3 n +1 ;(2)x n =(1? 1 2 )(1 ? 1 4 )...(1 ? 1 2 n ); (3)x n =1+ 1 2! + 1 3! + ... + 1 n! 4x n = sin1 1 2 + sin2 2 2 + ... + sinn n 2 ; (5)x n = a 0 + a 1 q + a 2 q 2 + ... + a n q n (|q| < 1,|a k |≤M, k =1,2, ...). 10A0BEB5DEAQCJAQCHAJA9ASBLCXDGAC (1) x 1 = √ 2,x n+1 = √ 2+x n ;(2)0<x 1 < 1,x n+1 =1? √ 1 ? x n ; (3)0 <x 1 < √ 3,x n+1 = 3(1 + x n ) 3+x n ;(4x 1 > 0,a>0,x n+1 = 1 2 (x n + a x n ); 3 (5) a 1 =1,a n =1+ 1 1+a n?1 ;(6)x 1 = √ 2,x n+1 = √ 2x n ; (7) A>0, 0 <x 1 < 1 A ,x n+1 = x n (2 ? Ax n ); (8) x 1 =sina, x n+1 =sinx n . 11A0C4{a n }BFBOBAA9{b n }BFBODBA9 lim n→∞ (b n ? a n )=0A2BEB5AC {a n }B0 {b n }BQCHAJA9ASBHAX DICVCXDGA2 12A0C40 <a 1 <b 1 ,AUa n+1 = √ a n b n ,b n+1 = a n + b n 2 ,n=1,2,3, ...BLBEAC{a n }B0 {b n }CXDGBD B7BHDIBKA2 13A0C40 <a 1 <b 1 <c 1 ,AUa n+1 = 3 1 a n + 1 b n + 1 c n ,b n+1 = 3 √ a n b n c n ,c n+1 = a n + b n + c n 3 ,n= 1,2,3, ...A2BLBEAC {a n }A0{b n }A0{c n }CXDGBDB7BHDIBKA2 14A0C4x 1 = a,x 2 = b,x n = x n?1 + x n?2 2 (n ≥ 3),AGAUALBNDACRBP?BEB5{x n }CHAJASBLBDCXDGA2 15A0BEB5AC A51A7BWCJAQ{x n }CHAJBHx n > 0(n =1,2, ...)A9B9 lim n→∞ n √ x 1 x 2 ...x n = lim n→∞ n √ x n . A52A7BW x n > 0(n =1,2, ...)CHAJBH lim n→∞ x n+1 x n BDB7A9B9 lim n→∞ n √ x n = lim n→∞ x n+1 x n . A53A7 lim n→∞ n n √ n! = e. 16A0BW lim n→∞ a n = a, lim n→∞ b n = b A2BEB5 A51A7 lim n→∞ a 1 + a 2 + ... + a n n = a; A52A7 lim n→∞ a 1 b n + a 2 b n?1 + ... + a n b 1 n = ab. 17A0D4 a n = braceleftBig (1 + 1 n ) n bracerightBig ,b n = braceleftBig (1 + 1 n ) n+1 bracerightBig ,n∈ N + . BEB5AC (1)CJAQ{a n }BFBOBAD7AXDIAD (2)CJAQ{b n }BODBC2AXDIAD (3A7CJAQ{a n }CECVAKAQAD (4)D4 lim n→∞ (1 + 1 n ) n = e. BLBEAC lim n→∞ (1 + 1 + 1 2! + ... + 1 n! )=e; (5) 1 (n +1)! <e? (1 + 1 + 1 2! + ... + 1 n! ) < 1 n!n ; (6) e =1+1+ 1 2! + ... + 1 n! + θ n n!n ( n n +1 <θ n < 1); (7) eD3D8?CJA2 BRCNAKDBCT 18A0C4AXAMBRA4B8C5BJCYBPA9ALCAB6BGCQB1BAA2BCBMBXCAB6A9CAA9B0CAB6AWAMBRA3CYA9BHA4C5BJ B0BRA3CYALB7B8C5ALCAB6BGCQB1BAA9BMBXCAB6A9CAB0B6C5AMBRA3CYA2D8BPB8C5BJCYBPA8D8CND1A9A5 1A7 D7AMBACQCEAXD1BRCYBPAFA52A7D7nCAB6CQAXBTC2BRCYBPAFA53A7BWnCAB6BGCQAXF n BRCYBPA9CGBL lim n→∞ F n F n+1 A5CTBJCPDDBJAMBRCYBPA8CEBAA7ASA6BJA7A2 4 CMB5ACC5D7CTCEAQBEAGCJACD6 Fibonacci AZC6BXCDD5B7(1202BA) AFA3CYBPBYBICJB3BJBJCJAMAO CRCIAVC7CSB8ADBJA9BDBJBJCJAQF n AJCQBSAZD3FibonacciCJAQAAAXBPBJCEA9CXDG?0.618BDCE”CTDJ C2C8”CJA9B7AVABBXCZA9BTATB2BIBHCOBTA4BJATAUA2 19AACPD5BLD2B6A5CEB7AFA3CFAXA2D2BJAMBK?CSDFA0BJCSB8ADBJA9DFAPBMBUBJAWAMB0D9C9BGDABJ CHDCD3AHADCMB5AABWBQBABMBUBJAWAMCDCJAZBLA9CQBJD9C9A4CUDEBLADD9C9DEBLCUC9A0BAAJBMBBDBC2A9C9 BMBUBJAWAMCDAUATBLA9AZCEBUD9C1AIADD9C9C1AIAYC9B5BABMBUAWAMBAD7A9B8B1A4BJCDCJAZBLA9BVBB?CS DEBQA2C4x n D3BMnBABJBMBUAWAMA9y n D3BDD9C9A9AWAZBGBABJAWAMBRBPB5BAD9C9A9CPAP y n = f(x n )A9 AZD3A8BLCKCJA9BVBMnBABJD9C9AYA7BPBMn +1BABJAWAMA9CGx n+1 = g(y n )A9AZD3CDATCKCJAAAWA1CH DCB2DFB8BVDECJB3AC x 1 → y 1 → x 2 → y 2 → x 3 → y 3 → x 4 → .... B7BCB3BHDFBUAPDCBJB4B8DEB3BJBNAQAC P 1 (x 1 ,y 1 ),P 2 (x 2 ,y 1 ),P 3 (x 2 ,y 2 ),P 4 (x 3 ,y 2 ), ......, P 2k?1 (x k ,y k ),P 2k (x k+1 ,y k )(k =1,2,3, ...), BDBJCPAXBJP 2k BQAYBSx = g(y),P 2k?1 AYBSy = f(x)A9BVCXCPCCAAAWAZBCBKCHDCCODJAMCABLD2A9CPAP AZD3BLD2B6A5A2 A6CWD3A9B7B0CF 1991BABMBUAWAMD330CZBSA9BUD9D3 6B4 /kgAD 1992BABMBUAWAMD325 CZBSA9 BUD9D3D38B4ABkgAAAOBF1993BABJBMBUAWAMD328CZBSAABWD4B4B8BGBJA2C1CLBCCLC5AWB6CBA9ASD8 BPBMBUBGBABJD9C9B0BGBABJAWAMBGDAA0ADBABJAWAMB0BGBABJD9C9BGDABQCEDHA6CHDCAA (1)CGBRBPA8BLCKCJ y n = f(x n )CLCDATCKCJ x n+1 = g(y n ); (2)BL lim n→∞ x n+1 B0 lim n→∞ y n+1 ; (3)D7BWC6BACQBMBUBJAWAMB0D9C9CEC4CUBMAZD6BPAFBWB9CFD6BPA9BLB8D6BPBJAWAMCLD9C9A2 CMB5ACAKBOAODBCTABBQD0B2C0A0AXBFBUBNANA3CCAACJACC2D9CVB9A4A5C7BKDGB3B8AHC3A91998BAA7A9 APCZAIA1BEACCJACDCA3CJACC2D9DBCTCYA4A5ARB5BOA0C0BEBIA0AFBDB4A0ANABBSANA9C7BKDGB3B8AHC3A9 1986BAA7AA