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ลําดะบแลัอนุกรม       (Sequences and Series)   หนะงสือเรียนออนไลน ชวงชะนที่ 4                             ้ชุด “คณิตศาสต...
F F       ˈ        F         10                           15 F                                              F             ...
1                                                        1 24     1.1                                                    1...
11.1                  1.1              (sequence)  ˆ กF                                   ก ˈ                             ...
F           1.3 ก                           F an + 2 =          1                        ก                   กn>2         ...
4) f2 : {1, 3, 5, } → R          5) g2 : {1, 2, 3, } → S2          6) h2 : S3 → R2.                    F      ก       FF  ...
5 = 3+2                               7 = 5+2                                     F                            { 1, 1, 3, ...
ก                               F              ก                                     F                  1.3              {...
a1   =   a1              a2   =   ra1              a3   =   ra2 = r2a1              a4   =   ra3 = r3a1              an = ...
1.4                                   an = n n 1                                          +              ก               F...
n ˈ                            ก                   F           ก                F F F F F2                                ...
F           1.10                              F       lim n n 1 = 1                                                       ...
F                ˈ            ก ก                                           F ก ก         F กF ˆก ก ก               1.2 ก ...
lim ( a n + b n ) = L + M                  n→∞4)              an bn = an + ( bn)      ก F 3) ก               F      F ก5) ...
ก lim a n = L                            F          F       lim a n = |L|                                       n→∞       ...
                                                      =  lim 1k′  ⋅  lim 1  (   1.2 F 5))                         ...
                                                          = m ⋅ ℓn  lim a n                                           ...
2)            n→∞                lim b n                                  2                                     (         ...
F                                                      F           F                     F                        1.4     ...
F            1.13                       F                       (   2 3        4                                          ...
L ε < an ≤ b n     b n ≤ cn < L + ε                        F F L ε < bn < L + ε                       ก |bn L| < ε        ...
ʿก       1.41.                     F F                                   1     1)                lim        nn            ...
6.            ก                  F an ˈ        F            FF F   lim a n = 0         F   lim a n = 0                    ...
F   1.18                                         F                  ก                      FF                             ...
ʿก   1.5            F            ก      FF   F F          F ก F F F1.   an = ( −1) n nn                        2          ...
1.6                      ʾ            ก                                                             ˈ           F กก      ...
an + 1                    xn           =   an       -----(1.8.1)                                   a +a                   ...
2                                                                          ก2.1            ก             2.1         ก (se...
n                       2)     ∑ ka i = ka1 + ka2 + ka3 +                           + kan                             i=1 ...
S4 = S3 + 1 = 15              8    8                = 1+ 87               1   31    S5 = S4 + 16 = 16                     ...
ʿก       2.11.                           กn                   FF               F           n           n(n + 1)     1)    ...
2.2            ก                    2.5       ก                       (arithmetic series)          ก       F กก           ...
= k 2 1 [2a1 + kd]                                                 +                                             = k 2 1 [...
F   2.6         F log93, log9(3x 2), log9(3x + 16) ˈ          F ก    ก                      ก            S ˈ              ...
FF       217 = n [ 2(7)+ (n − 1)8 ]                                    2                                  = n ( 6 + 8n )  ...
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
Lecture 010 sequence-series ลำดับและอนุกรม
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Lecture 010 sequence-series ลำดับและอนุกรม

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Lecture 010 sequence-series ลำดับและอนุกรม

  1. 1. ลําดะบแลัอนุกรม (Sequences and Series) หนะงสือเรียนออนไลน ชวงชะนที่ 4 ้ชุด “คณิตศาสตรบนเว็บไซต” เลมที่ 10 สะทธา หาญวงศฤทธิ์ F F F F . . 2537 F F F ก F ก F F ก ก F
  2. 2. F F ˈ F 10 15 F F F ก ก ก F ก 1 F กFก ʾ ก 2 F ก F กก F ก ก F ก ก F Fก F F ˆ ก ก ก F ก F F F ʾ F ก ก F ʽ F 3 ก ก ก F ก F กก F ก F ก ก F F ก F ก ก ก F ก F ก F F F F F ก ก ก ก ก ก F ก F F F Fก F F F กF F F F F กก F F F F F ก กF ก FFF F F F F กF F F กF F F 4 . . 2549 ก 1 ก F F ˈ ก F F ก F1.5 กFก F Fก F F ʿก F F FF F Fก F FFF ก F F F ก F ก F 16 ก . . 2549
  3. 3. 1 1 24 1.1 1 1.2 3 1.3 5 1.4 7 1.5 กFก 20 1.6 ʾ ก 232 ก 25 36 2.1 ก 25 2.2 ก 29 2.3 ก 333 ก F ก 37 44 3.1 ก F ก 37 3.2 ก F F ก ก F ก 41 3.3 ก ก ˈ ก ก F ก 43 ก F F F 45
  4. 4. 11.1 1.1 (sequence) ˆ กF ก ˈ ˈ F F ก F {an} F an n ˈ ก F 1.1 ก F an = 3n + 1 5 F ก n = 1, 2, 3, 4, 5 FF a1 = 3(1) + 1 = 4 a2 = 3(2) + 1 = 7 a3 = 3(3) + 1 = 10 a4 = 3(4) + 1 = 13 a5 = 3(5) + 1 = 16 5 F ก an = 3n + 1 4, 7, 10, 13, 16 F 1.2 ก F { 3, 2 , 4 , 8 , 16 } ˈ 3 3 3 3 F ก 3 = ( 1) 30 ( ) 2 3 2 = ( 1) ( ) 2 3 21 3 4 = ( 1) ( ) 3 3 22 3 8 = ( 1) ( ) 4 3 23 3 16 = ( 1) ( ) 5 3 24 F an = ( 1)n ( ) 2 3 n −1 n = 1, 2, 3, 4, 5
  5. 5. F 1.3 ก F an + 2 = 1 ก กn>2 5 F ก 2 n−1 F m = n+2 n=m 2 m=2 Fn=0 an + 2 = a m = 1 = 1 2 (m − 2) − 1 2 m−3 Fก Fm=n F F an = 1 ก กn>3 2 n−3 n = 4; a4 = 1 = 1 2 4−3 2 n = 5; a5 = 1 = 1 2 5−3 2 2 n = 6; a6 = 1 = 1 2 6−3 2 3 n = 7; a6 = 1 = 1 = 1 2 7−3 2 4 4 n = 8; a8 = 1 = 1 2 8−3 2 5 5 F ก {1 , 1 , 1 , 1 , 1 } 2 2 2 2 3 4 2 5 F ก n F F n=1 F 1.3 F n=4 ก F FF ก ก ˈ 2 F ก F F F ก ก ก F ก (finite sequence) F F F ก ก กF F ก (infinite sequence) F F F Fก F ก F Fก F F ก ˆ กF ก ˈ ˈ F F ก ˆ กF ก ˈ ˈ Fก F กก F ก ก Fก ก F F F F F ก F F F ก ก F F F ʿก 1.11. F ˆ กF ก FF ˈ F FR ˈ ก Si ก F ˈ 1) f1 : {1, 2, 3, 4} → R 2) g1 : {1, 2, 3, } → R 3) h1 : {1, 3, 5, } → S12 ก
  6. 6. 4) f2 : {1, 3, 5, } → R 5) g2 : {1, 2, 3, } → S2 6) h2 : S3 → R2. F ก FF ˈ ก F ก 1) {an | an = n2 4 1 ≤ n ≤ 6, n ˈ ก} 2) {an | an = 23 n ≥ 1, n ˈ ก} n +1 3) {an | an = 1 n ≥ 1, n ˈ ก} 1− 1 n − n1 1 +1.2 1.2 (arithmetic sequence) F F F ก F ก F F F F (common difference) ก 1.2 FF a1 = a1 a2 = a1 + d a3 = a2 + d = (a1 + d) + d = a1 + 2d a4 = a3 + d = (a1 + 2d) + d = a1 + 3d a5 = a4 + d = (a1 + 3d) + d = a1 + 4d an = a1 + (n 1)d กn F an = a1 + (n 1)da1 F ก d F F an F n F 1.3 F { 1, 1, 3, 5, 7, } ก 1= 1 1 = ( 1) + 2 3 = 1+2 F F 3
  7. 7. 5 = 3+2 7 = 5+2 F { 1, 1, 3, 5, 7, } an = 1 + (n 1)(2) = 3 + 2n F 1.4 F {1, 3, 5, 7, } ก 1 = 1 3 = 1+2 5 = 3+2 7 = 5+2 F {1, 3, 5, 7, } an = 1 + (n 1)(2) = 1 + 2n F 1.5 F ก, Fก F F ˈ F F FF Fก Fก 14 26 F F Fก FF Fก F ก Fก n 2 FF Fก ก F กF n 1 2 n+1 2 ก ก an = a1 + (n 1)d -----(1.2.1) F 14 = a(n/2) + 1 = a1 + ( n + 1 ) − 1 d = a1 + ( n ) d -----(1.2.2)  2  2 F 26 = a(n/2) 1 = a1 + ( n − 1 ) − 1 d = a1 + ( n − 2 ) d  2  2 = a1 + ( n ) d 2d -----(1.2.3) 2 F ก (1.2.2) ก (1.2.3) FF 26 = 14 2d 2d = 14 + 26 = 12 d=6 F d=6 ก (1.2.2) FF 14 = a1 + ( n ) (6) = a1 + 3n 2 a1 = 14 3n -----(1.2.4) ก ก an = a1 + (n 1)d F a1 = 14 3n, d = 6 FF an = ( 14 3n) + (n 1)(6) = 14 3n + 6n 6 = 20 3n F an = 20 3n4 ก
  8. 8. ก F ก F 1.3 {a1, a2, a3, , an, } ˈ F กก F { a1 , a1 , a1 , , a1 , } ˈ 1 2 3 n F 1.6 F {1, 1 , 1 , 1 , } ˈ 2 3 4 F ก ก 1= 1 1 1 2= 1 () 2 1 3= 1 () 3 1 an = ( a1 ) n F {1, 2, 3, , an} ˈ {1, 1 , 1 , 1 , } ˈ 2 3 4 F ก F ก ʿก 1.21. F 5, x, 20, ˈ ก 12 F ก ˈ a 5, y, 20, ˈ F 6 ˈ b y<0 F a+b F F2. F ก ˈ 200, 182, 164, 146, F F ก F 10 F ก F1.3 1.4 (geometric sequence) F F F ก Fก F ก F F F (common ratio) ก 1.4 FF F F 5
  9. 9. a1 = a1 a2 = ra1 a3 = ra2 = r2a1 a4 = ra3 = r3a1 an = ran 1 = rn 1a1 กn F an = a1rn 1 a1 F ก r F F an F n F 1.7 F {1, 1 , 1 , 1 , 16 , } 2 4 8 1 ก 1 = (1) 0 2 (2) 1 = 1 1 2 1 = 1 2 4 2 () 1 = 1 3 8 2 () n−1 F an = ( 1 ) 2 ก กn ʿก 1.31. ก F a, b, c ˈ 3 F ก ˈ 27 F a, b + 3, c + 2 ˈ F ก F a+b+c F Fก F2. ก F a + 3, a, a 2 ˈ F ก F F ˈ r F F ∞ n −1 ∑ ar Fก F n=13. F x, y, z, w ˈ F4 F ก x ˈ F ก F y+z = 6 z + w = 12 F F F 56 ก
  10. 10. 1.4 an = n n 1 + ก Fก ˈ F n F F ก ˈ F an n ก F 1.1 ก an = n n 1 + F n F ก F ก an = n n 1 + F ก F F y=1 ก F ก F F an F F F 1 an Fก 1 FF ก F lim a n = 1n→∞ ก L ก F F an F F F L FF ก F lim a n = L ก an F F F (convergent sequence) ก F Ln→∞ an F F F L ก an F F ก (divergent sequence) 1.5 ก F F an ˈ F Fก F an F F F L ก F F an ˈ F ก ก F an F F F F F 1.8 ก F an = n2n 1 + กn F ก F ˈ F F F ก F ˈ F F ก an = n2n 1 + Fn ˈ ก F F Fก ˈ F an F F F 7
  11. 11. n ˈ ก F ก F F F F F2 ˈ F F FF lim a n = lim ( n2n 1 ) = 2 + n→∞ n→∞ F 1.9 ก F an = n กn F an ˈ F F F ก F ˈ F F ก an = n กn Fก ˈ F n F Fก ˈ F an F F F ก F F F F F ก ก ก F ก FกF F F ก F (Real Analysis) กF 1.6 ก F ε> 0 กN F n≥N( N Fก ε) ก ก n F |an L| < ε F ก ก F 1.6 ก F F ก ก F ก ก Fก F F ก F |an L| < ε F กN F ก FFF F F F F F F 1.6 ก8 ก
  12. 12. F 1.10 F lim n n 1 = 1 + n→∞ Fε>0 |an L| = n n 1 − 1 = n n 1 − n + 1 = − n 1 1 = n 1 1 < ε + + n+1 + + F F 1 < ε(n + 1) (‹ ε > 0) < εn + ε 1 ε < εn 1−ε < n ε ε 1<n 1 ก ε> 0 F F 0 < ε <1 1 ε 1<0<n 1 กN≥ ε 1 1 FF lim n n 1 = 1 + F ก n→∞ F ˈ ก ก F F F ก F F FFF F ˈ ʿก F ก F F F F ก ก ก F 1.6 ก F F F F 1.1 (Uniqueness of limit of sequence) F lim a n = L1 lim a n = L2 F F F L1 = L2 n→∞ n→∞ F ก F ε> 0 ก lim a n = L1 lim a n = L2 FF ก N1, N2 n→∞ n→∞ ก F F |an L1| < ε กn≥N 2 |an L2| < ε 2 |(an L1) (an L2)| ≤ |an L1| + |an L2| (‹ ก ) ≤ ε+ε 2 2 = ε F |(an L1) (an L2)| = | (L1 L2)| = |L1 L2| = ε F F L1 = L2 F กF ก 1.1 ก F ก F F F F F F F ( F F F F F ) ก 1.1 FFF FF กε F |L1 L2| = ε F L1 = L2 F F 9
  13. 13. F ˈ ก ก F ก ก F กF ˆก ก ก 1.2 ก F lim a n = L, lim b n = M k ˈ FF n→∞ n→∞ 1) lim k = k n→∞ 2) lim ka n = kL n→∞ 3) lim ( a n + b n ) = L + M n→∞ 4) lim ( a n − b n ) = L M n→∞ 5) lim ( a n ⋅ b n ) = L ⋅ M n→∞ 6) lim n→∞ ( ab ) = M n n L 7) lim a n = lim a n = |L| n→∞ x→∞ F ก F ε> 0 N1, N2 ˈ ก 1) ก |k k| < ε F0<ε lim k = k F ก n→∞ 2) ก k=0 FF F F ก F ˈ F k≠0 ก lim a n = L กN n≥N F |an L| < ε k n→∞ |kan kL| = |k(an L)| = |k||an L| < |k|⋅ ε = ε k 3) ก lim a n = L, lim b n = M n→∞ n→∞ ก N1, N2 n ≥ max{N1, N2} F |an L| < ε 2 |bn L| < ε2 |(an + bn) (L + M)| = |(an L) + (bn M)| ≤ |an L| + |bn L| < ε+ε =ε 2 210 ก
  14. 14. lim ( a n + b n ) = L + M n→∞4) an bn = an + ( bn) ก F 3) ก F F ก5) ก Fα ˈ |an| ก lim a n = L, lim b n = M n→∞ n→∞ ก N1, N2 n ≥ max{N1, N2} F |an L| < 2( Mε+ 1 ) |bn M| < 2ε α |anbn LM| = |anbn LM anM + anM| = |(anbn anM) + (anM LM)| = |an(bn M) + M(an L)| ≤ |an(bn M)| + |M(an L)| = |an||(bn M)| + |M||(an L)| < α⋅ 2ε + 2( Mε+ 1 ) α < ε+ε =ε 2 2 lim ( a n ⋅ b n ) = L ⋅ M n→∞ an6) b n = an ⋅ b n bn ≠ 0 ก F 5) ก F F ก 17) ก lim a n = L กN n≥N F |an L| < ε n→∞ FF ε < an L < ε L ε < an < L + ε F F an < L + ε |an| < |L + ε| ≤ |L| + |ε| F F |an| |L| < |ε| a n − L < ε = ε (‹ ε > 0) lim a n = |L| n→∞ ก |an L| < ε F F a n − L < |ε| = ε (‹ ε > 0) lim a n = lim a n n→∞ x→∞ F F 11
  15. 15. ก lim a n = L F F lim a n = |L| n→∞ x→∞ ก F max{N1, N2} ก N1, N2 ก F N = max{N1, N2} N ≥ N1 N ≥ N2 กก ก ก F ก ก F F ก ก F ก ก FF F 1.3 1) lim 1k = 0 ก กk n→∞ n 2) lim n k F ก n→∞ 3) lim m = 0 k ก m, k k>0 n→∞ n 0 1<x<1 1 x=1 4) lim x n = n→∞ F ก x>1 5) F lim a n = L ma n ˈ ก กn F n→∞ lim ( m a n ) = m L n→∞ F 1) F F ก F ก F P(k) F lim 1k = 0 ก กk n→∞ n : k=1 F F lim 1 = 0 n ˈ n→∞ :ก F k′ ˈ ก F lim 1k′ = 0 n→∞ n lim 1 = lim 1 k′ + 1 k′ n→∞ n n→∞ n ⋅ n = lim n→∞ n 1 1 ( ⋅) k′ n12 ก
  16. 16.     =  lim 1k′  ⋅  lim 1  ( 1.2 F 5))  n→∞ n   n→∞ n  = 0⋅0 ( ก ) = 0 ก F F F lim 1k = 0 ก กk n→∞ n 12) ก nk = ( n1 ) k 1 lim n k = lim n→∞ n→∞ 1 ( ) nk lim 1 n →∞ ( ) = lim 1 n →∞ nk lim 1 ก F 1) lim 1k = 0 F n →∞ ก ก F F F n→∞ n ( ) lim 1k n →∞ n lim n k F F F F F ก n→∞3) m = m⋅ 1 ก F 1) ก F F ก nk nk4) ก 1 < x < 1: ก xn ˈ |x| < 1 ˈ F F ก F 1) ก F F ก ก x = 1: F F ก F ˈ ก x > 1: ก xn ˈ |x| > 1 ˈ F ก 15) F L′ = lim ( m a n ) = lim ( a n ) m n→∞ n→∞  1 FF ℓn L′ = ℓn  lim ( a n ) m   n→∞  = lim ℓn ( a n ) m  1 n→∞     = lim  m ⋅ ℓn ( a n )   1  n→∞ = lim ( m ) ⋅ lim ℓn ( a n )  1   n→∞ n→∞ F F 13
  17. 17.   = m ⋅ ℓn  lim a n  1  n→∞  = m ⋅ℓn L 1 1 = ℓ nLm 1 L′ = Lm = m L ก ˆ กF ก ˈ ˆ กF F ก   lim [ ℓ na n ] = ℓ n  lim a n  F an n→∞  n→∞  F 1.11 ก FF (F ) 1) 2n + 1 an = 3n + 4 2 2) bn = 3n 2 − 4 2n + 1 3) an + b n 4) an ⋅ b n an 5) bn 1) lim a n = lim ( 3n + 4 ) 2n + 1 n→∞ n→∞  2+ 1  n = lim  4  n→∞  3+ n     1  lim 2  +  lim n  =  n→∞   n→∞     4  lim 3  +  lim n   n→∞   n→∞     1  lim 2  +  lim n  =  n→∞   n→∞     1  lim 3  + 4⋅ lim n   n→∞   n→∞  = 2+0 3 + 4⋅0 = 2 314 ก
  18. 18. 2) n→∞ lim b n 2 ( = lim 3n 2 − 4 n→∞ 2n + 1 )  3 − 42  = lim  n1  n→∞  2 + n 2       4   lim 3  −  lim 2  =  n→∞   n→∞  n    1   lim 2  +  lim 2   n→∞   n→∞ n  = 32 − 0 + 0 3 = 23) lim ( a n + b n ) = lim a n + lim b n n→∞ n→∞ n→∞ = 2+23 3 = 13 64) lim ( a n ⋅ b n ) = lim a n ⋅ lim b n n→∞ n→∞ n→∞ = 2 ⋅2 3 3 = 1 lim a n5) lim n→∞ ( ) an bn = n →∞ lim b n n →∞ 2 = 3 3 2 = 4 9 F F 15
  19. 19. F F F F 1.4 a 0 + a1n + a 2 n 2 + a 3 n 3 + ... + a s−1x s−1 + a s x s ก F Pn = n ˈ ก s, t ˈ b 0 + b1n + b 2 n 2 + b 3 n 3 + ... + b t−1x t −1 + b t x t ˈ F 1) F s<t F lim Pn = 0 n→∞ a 2) F s=t F lim Pn = bs n→∞ t 3) F s>t F Pn F ก 1.4 F F FFF F F ˆ F F F F 1.4 ก a F 1.12 ก F an = 2 + 3n + n2 bn = 1 3n + 3n2 n3 Pn = b n ก n กn≥2 F lim Pn n→∞ a Fก F lim Pn = lim bn n→∞ n→∞ n = lim ( 2 + 3n + n 2 2 3 n→∞ 1 − 3n + 3n − n )  n3 2 + 3 + 1 = lim  n3 n2 n2 ( )  ( n→∞  n 3 13 − 32 + n − 1  n n 3 )   23 + 32 + 12  = lim  1 n 3n 3n  n→∞  n 3 − n 2 + n − 1    0+0+0 = 0−0+0−1 = 0 F 1.4 ก F F F s = 2, t = 3 s< t ก F 1) F F lim Pn = 0 n→∞16 ก
  20. 20. F 1.13 F ( 2 3 4 lim 4 + 3n2 − n3 + 2n4 n→∞ 3 − n + n − 3n ) ก F กs=t=1 ก 1.4 F 2) FF n→∞ (2 3 4 lim 4 + 3n2 − n3 + 2n4 = − 2 3 − n + n − 3n 3 ) F ก F 1.13 FFF F F F 1.4 F F F F Fก F  n1 − n1  3 2 F 1.14 F lim  1  n→∞  n − n 2  ก s = 1, t = 1 2 F s<t  n1 − n1  3 2 1.4 F 1) F F lim  1 = 0 n→∞  n − n 2  F ก ก F F 1.14 F F ก ก F 1.4 F Fก ก F ก ก F ก F F F F ก F ก Fก F F ก F F F F ก ก F F 1.5 (Squeeze Theorem for sequence) ก F an, bn, cn ˈ an ≤ b n ≤ cn ก กn F F F ˈ 1) F lim a n = L lim c n = L F lim b n = L n→∞ n→∞ n→∞ 2) F b n F ก F cn F ก F F ก F an, bn, cn ˈ an ≤ b n ≤ cn ก กn 1) ก Fε>0 F lim a n = L lim c n = L n→∞ n→∞ ก N1, N2 ก n ≥ max{N1, N2} F |an L| < ε |cn L| < ε F F L ε < an cn L < ε ก cn < L + ε F F 17
  21. 21. L ε < an ≤ b n b n ≤ cn < L + ε F F L ε < bn < L + ε ก |bn L| < ε F F lim b n = L n→∞ 2) F F (contraposition) F F ก F ˈ F F cn F F F b n F F F Fε>0 lim c n = L n→∞ กN กn≥N F |cn L| < ε F F L ε < cn < L + ε ก b n ≤ cn ก กn F F L ε < bn < L + ε |bn L| < ε F F bn F F ก F F F F F ก F ˈ 2 F 1.15 F bn = 2n 3 + 5 กn F F F ก 5n + 4 2 cn = 2n 3 = 5n 2 5n 2 2 an = 2n3 + 5 = 2n 3 + 5 5n + 10 ( 5 n +2 ) F an ≤ b n ≤ cn ก กn ก lim a n = lim c n = 0 n→∞ n→∞ 1.5 FF lim b n = 0 n→∞ F 1.16 F lim nn n→∞ 2 F nn = n −1 + 1n n 1 < n 2 2 2 2n 2n F nn < 1 n F 1n < nn < 1 n 2 2 2 ก lim 1n = 0 = lim 1 n n→∞ 2 n→∞ 1.5 F F lim nn = 0 n→∞ 218 ก
  22. 22. ʿก 1.41. F F 1 1) lim nn n→∞ lim ( 0.999... + 1 ) n 2) n n→∞ ( ) 1 n 3) lim 1 + 2 1 n→∞ n + 3n + 2  1 1 1 1  2n 4) lim  n 2 + n 4 + n 8 + ... + n 1 1 1  n→∞  n + n 3 + n 5 + ... + n 2n-1    5) lim ( ln1n ) n n→∞ 1 6) lim ( ) 1 n ln n n→∞2. F 1.5 FF lim sin n = 0 n lim cos n = 0 n F ก n→∞ n→∞ FF 1 1) n→∞ lim ( ) sin n n n lim ( sin n ) n 2) n n→∞ 1 3) lim ( cos n ) n n n→∞ lim ( cos n ) n 4) n n→∞  1 n n 3. กn ก F Mn =   an = det(Mn) F F − n n + 1 1   lim a n n→∞4. F กn≥4ก F an = n4 + 1 F lim a n 13 + 2 3 + 33 + ... + n 3 n→∞ 2 n n5. F an = n +2n + 1 bn = 2 n− 5 F F n ˈ an bn + anbn F F 3n + 1 5 +9 F F 19
  23. 23. 6. ก F an ˈ F FF F lim a n = 0 F lim a n = 0 n→∞ n→∞1.5 กFก {1, 2, 3, 4, 5, 6, } F ก F ก F F F F F an = ( 1)nn ก กn ก F {1, 3, 5, 7, 9, 11, } ก F F F F กF bn = ( 1) (2n 1) n ก กn FF F F ก F F F F กก F F F F ˈ ( 1)nAn An ˈ F ˈ F ก F ก ก F F ก F ก (oscillating sequence) 1.7 ก F ก (oscillating sequence) F an = ( 1)nAn ก ก n An ˈ F F ก F 1.17 F ก FF ˈ กFก 1) an = 3n − 31 4n + 2) bn = ( 1)n 3n − 31 4n + 3) cn = ( 1)n(3n 1) 1.7 1) an F ก F F ( 1)n F ˈ กFก 2) bn ก F ( 1)n F lim 3n − 31 = 4 4n + 3 F ˈ F F n→∞ bn F ˈ กFก 3) cn ก F ( 1)n An = 3n 1 ˈ F ก cn ˈ กFก20 ก
  24. 24. F 1.18 F ก FF ˈ กFก 1) π an = sin n4 2) π bn = sin ( 1)n n4 3) cn = cos ( 1)n nπ n 2 F F ก F ˈ กFก ก an F ก F ( 1)n F FF bn cn ก F ( 1)n F Fก F F F ( 1)nAn ก F π F ( 1)n n4 ก F ( 1)n nπ ˈ n Fก F 2 ˆ กF F ˆ กF F FF 1.19 F an = cos ( 1)n nπ ˈ n F F F ก F ˈ F F F 2 ก F 1.18 F an = cos ( 1)n nπ n F F กFก 2 cos nπ n n ˈ F 2 ก cos ( 1)n nπ n = 2 cos nπ n n ˈ 2 ( )  1 ( ) ( ) ( ) ( )  2 4 6 8 ก lim cos nπ = lim 1 − 2! nπ + 4! nπ − 6! nπ + 8! nπ − ...  1 1 1 n→∞ 2n n→∞  2n 2n 2n 2n   1 ( ) ( ) ( ) ( ) nπ − 1 nπ + 1 nπ − ...  F F 2 4 6 8 F F lim  − 2! nπ + 4! n 1 6! 2 n 8! 2 n  ก n→∞  2 2n  lim cos nπ n→∞ n ( ) 2 = 1 ก FF n→∞ ( lim − cos nπ = ( 1) = 1 (‹ cos nπ = 1 n 2 ) n ˈ ˈ ) FF n→∞ n ( ) lim cos nπ = lim − cos nπ = 1 n→∞ n 2 ( 2 ) F lim ( −1) n cos nπ  = 1 n ˈ ก n→∞  2n  an = cos ( 1)n nπ ˈ F F 2n F F 21
  25. 25. ʿก 1.5 F ก FF F F F ก F F F1. an = ( −1) n nn 2 22. an = ( −1) n n n 2 −n3. an = ( −1) n e 3 n n sin n4. an = ( −1) 2 n n n5. an = ( −1) ln n22 ก
  26. 26. 1.6 ʾ ก ˈ F กก ก F กF ʾ ก (Fibonacci Sequence) ก F ก F ʾ ก กก ก F 1, 1, 2, 3, 5, 8, 13, FFF ก F ก F F F 3 ˈ F F Fก กก ก F กF F 2 F F F 2 = 1+1 = a 1 + a2 3 = 1+2 = a 2 + a3 5 = 2+3 = a 3 + a4 8 = 3+5 = a 4 + a5 FF ก กn≥3 F F F F F an = a n 2 + a n 1 ก F F (initial value) a1 = 1 a2 = 1 1.8 ʾ ก (Fibonacci Sequence) F F an = a n 2 + a n 1 a1 = 1 a2 = 1 ก กn≥3 F 1.20 F 6, 7, 13, 20, 33, ˈ ʾ ก F ก 13 = 6 + 7 20 = 7 + 13 33 = 13 + 20 FF F F an = a n 2 + a n 1 FF F F F1 F ก F F F ʾ ก 1, 1, 2, 3, 5, 8, 13, F ก F n F n+1 กn 1 , 2 , 3 , 5 , 8 , 13 , 1 1 2 3 5 8 a Fก F x1 = 1 , x2 = 1 , x3 = 2 , 1 2 3 F F xn F xn = na + 1 n F F 23
  27. 27. an + 1 xn = an -----(1.8.1) a +a = n a n −1 ( ก F n = n+1 ก an = an 2 + an 1) n a n a n −1 = a + a n n 1 = 1+ a n a n −1 = 1+ x1 ( ก F n=n 1 ก 1.8.1) n −1 F xn F F F F F xn 1 ก F F F ก FF lim x n = lim 1 + x 1 n→∞ n −1 n→∞ ( ) x = 1+ 1 x x 1= 1 x x2 − 1 = 1 x x2 x 1 = 0 -----(1.8.2) กF ก (1.8.2) FF ก ˈ ก x = 1 + 25 2 F ˈ F กก F กก ก F ก F F (golden ratio) F F F F 1) F ก F F F ก F F 2) F F F ก F ˈ F 3) F F F ก กF ˈ F ˈ F ก F F F F (golden rectangle) 4) F F Fก F ก ˈ F ʿก 1.61. F n ก F F ʾ ก 1) n = 9 2) n = 13 3) n = 162. F ก กก F 100 ʾ ก ก24 ก
  28. 28. 2 ก2.1 ก 2.1 ก (series) ก ก F F F ก ก F Σ ก 1.1 F ก F F Σ F 2.2 n กn F a1, a2, a3, , ai ˈ F ∑ a i = a1 + a2 + a3 + + ai i=1 Σ F ก F F F F 2.1 n 1) ก ก i F ai = k F ∑ a i = nk i=1 n n 2) ∑ ka i = k ∑ a i i=1 i=1 n n n 3) ∑ ( ai + bi ) = ∑ ai + ∑ bi i=1 i=1 i=1 n n n 4) ∑ ( ai − bi ) = ∑ ai ∑ bi i=1 i=1 i=1 F 1) ก F ai = k ก กi FF n ∑ a i = a1 + a2 + a3 + + an = k + k + k + + k = nk i=1 n
  29. 29. n 2) ∑ ka i = ka1 + ka2 + ka3 + + kan i=1 = k(a1 + a2 + a3 + + an) n = k ∑ ai i=1 n 3) ∑ ( a i + b i ) = (a1 + b1) + (a2 + b2) + (a3 + b3) + + (an + bn) i=1 = (a1 + a2 + a3 + + an) + (b1 + b2 + b3 + + bn) n n = ∑ ai + ∑ bi i=1 i=1 4) ai bi = ai + ( bi) ก F 1) ก F 3) ก F F ก ก 2 F กF ก F ก ก ก F ก F ก ก ˈ 2 F กF ก ก ก F ก F 2.3 1) ก ก (finite series) ก F ก 2) ก F ก (infinite series) ก F F ก F ก ก ก ก F F F กก F F ˈ ก F F (convergent series) ก F ก (divergent series) 2.4 1) ก F F ก กF F F 2) ก F ก ก กF F ก ก F F F F F F 2.3 2.4 ก F 2.1 1 + 1 + 1 + + 1024 ˈ F 2 4 ก 1 ก ก 2.3 FF ก ˈ ก ก F F ก F F F F Sn ˈ กF n F ก ก FF S1 = 1 = 1 S2 = 1 + 1 = 2 3 2 7 S3 = S2 + 1 = 4 3 = 1+ 4 426 ก
  30. 30. S4 = S3 + 1 = 15 8 8 = 1+ 87 1 31 S5 = S4 + 16 = 16 15 = 1 + 16 1 63 S6 = S5 + 32 = 32 31 = 1 + 32 S7 = S6 + 64 = 127 1 64 63 = 1 + 64 1 255 S8 = S7 + 128 = 128 = 1 + 127 128 1 511 S9 = S8 + 256 = 256 255 = 1 + 256 S10 = S9 + 512 = 1023 = 1 + 512 1 512 511 1 2047 1023 S11 = S10 + 1024 = 1024 = 1 + 1024 n −1 F F Sn = 1 + 2 n −−1 = 2 1 1 n −1 22 lim S n = lim 2 − n1−1 n→∞ n→∞ 2 ( )=2 2.4 FF ก ˈ ก F F ก ก F ก S11 F F F F ก ˈ ก ก (finite geometrical series) ก F ก 1024 = S11 2047F 2.2 F ก 1+ 1 + 1 + + 1 + 2 3 n ˈ ก ก 2.3 ก ˈ ก F ก F F ก F F F F Sn ˈ กF n F ก ก FF S1 = 1 S2 = 1 + 1 = 2 2 3 S3 = S2 + 1 = 11 3 6 = 2 1 6 25 S4 = S3 + 1 = 12 1 = 2 + 12 4 S5 = S4 + 1 = 137 5 60 = 2 + 17 60 S6 = S5 + 1 = 147 6 60 27 = 2 + 60 S7 = S6 + 1 = 1089 7 420 249 = 2 + 420 F F F กF F F ˈ ก F ก ก ก ก กF ก F ก 3 F F F 27
  31. 31. ʿก 2.11. กn FF F n n(n + 1) 1) ∑i = 2 i=1 n 2 n(n + 1)(2n + 1) 2) ∑i = 6 i=1 2 n 3  n  3) ∑i =  ∑ i i=1  i=1 2. ก กn F F F F n ∑ ( ai + bi ) 2 1) i=1 n ∑ ( ai + bi ) 3 2) i=1 n n 2 n 23. F F F ∑ ( ai + bi ) ≤ ∑ ai + ∑ bi 2 ก กn i=1 i=1 i=1 10 10 10 104. F ∑ x i = 8, ∑ y i = 4 ∑ ( 5 − x i )( y i + 2 ) = 76 F ∑ xiyi F Fก F i=1 i=1 i=1 i=15. ก ก ก FF 1) 1 3 + 5 7 + 9 + 99 2) 1 2+3 4+5 100 3) 1 1+2 3+5 8+ 55 4) 1 1 +1 1 + 2 4 8 ∞ 5) 1 ∑ (n + 3)(n + 4) n =16. F sin21° + sin22° sin23° + sin289°28 ก
  32. 32. 2.2 ก 2.5 ก (arithmetic series) ก F กก กก F 2.3 ก F an = 2n + 3 ˈ ก an 10 F ก 10 10 ∑ an = ∑ ( 2n + 3 ) i=1 i=1 10 10 = ∑ ( 2n ) + ∑ ( 3 ) i=1 i=1 10 = 2 ⋅ ∑ n + (10)(3) i=1 = 2 ⋅ 10 (10 + 1) + (10)(3) 2 = 110 + 30 = 140 ก F ก ก ก F F Fก F F F F 2.3ก F F ก F 2.2 กn กn F ก ก F ก F Sn F ก ก Sn = n [ 2a1 + (n − 1)d ] 2 a1 F ก ,d F F F F P(n) F Sn = n [ 2a1 + (n − 1)d ] 2 ก กn : Fn=1 F F S1 = 1 [ 2a1 + (1 − 1)d ] = a1 2 : Fn=k F P(k) ˈ F P(k + 1) ˈ F Sk + ak + 1 = k [ 2a1 + (k − 1)d ] + ak + 1 2 = k [ 2a1 + (k − 1)d ] + (a1 + kd) (‹ ak = a1 + (k 1)d) 2 2 = ka1 + k2 k d + (a1 + kd) 2 2 = (ka1 + a1) + k2 + k d 2 2 = 1 [2(k + 1)a1] + k2 + k d 2 2 = 1 [2(k + 1)a1 + k2 + kd] 2 = 1 [2(k + 1)a1 + k(k + 1)d] 2 F F 29
  33. 33. = k 2 1 [2a1 + kd] + = k 2 1 [2a1 + (k + 1 1)d] + = Sk + 1 P(k + 1) ˈ ก F F F Sn = n [ 2a1 + (n − 1)d ] 2 ก กn ก 2.1 Sn = n (a1 + an) 2 ก กn F ก 2.2 F F Sn = n [ 2a1 + (n − 1)d ] = n a1 + ( a1 + (n − 1)d )  2 2  F an = a1 + (n 1)d (‹ F ) Sn = n (a1 + an) 2 ก กn F 2.4 ก ก 1 +1 + 5 +1 + 4 3 12 2 +1 d1 = 1 3 1 = 1 4 12 5 d2 = 12 1 = 1 3 12 5 d3 = 1 12 = 12 1 2 dn = 121 F ก ก F ˈ ก F ก (a1) = 1 , d = 12 4 1 กF F F กF ก an = a1 + (n 1)d F a1 = 1 , d = 12 , an = 1 4 1 F F 1 = 1 + (n 1) 12 4 1 (n 1) 12 = 1 1 = 4 1 3 4 n 1 = 9 n = 10 ก 2.1 F F S8 = 10 ( 1 + 1) = 25 2 4 4 F 2.5 ก an = 10 2n ก 10 F ก ˈ n = 6; a6 = 10 2(6) = 2 n = 15; a15 = 10 2(15) = 20 ก 2.1 F F S10 = 10 (( 2) + ( 20)) = 110 230 ก
  34. 34. F 2.6 F log93, log9(3x 2), log9(3x + 16) ˈ F ก ก ก S ˈ ก F ก ก F 3S F F ก F Fก F log93, log9(3x 2), log9(3x + 16) ˈ F ก ก ก FF x d1 = log9(3x 2) log93 = log 9 3 − 2 3 ( ) d2 = log9(3x + 16) log9(3x 2) = log ( ) 3x + 16 9 3x − 2 F d1 = d2 (‹ F F ) x ( ) 3 x log 9 3 − 2 = log 9 3 x + 16 3 −2 ( ) 3x − 2 = 3x + 16 3 3x − 2 (3x 2)2 = 3(3x + 16) (3x)2 4(3x) + 4 = 3(3x) + 48 (3x)2 7(3x) 44 = 0 (3x 11)(3x + 4) = 0 3x 11 = 0 3x + 4 = 0 F 3x = 11 (‹ 3x + 4 = 0 F ˈ ) x = log311 ( ) F d1 = log 9 3 3 − 2 = log 9 ( 11 3 2 ) = log93 log 3 11 − ก S = 4  2 ( log 9 3 ) + (4 − 1) ( log 9 3 )  2  = 2  2 ( log 9 3 ) + 3( log 9 3 )    = 10 log93 = 10 ( 1 log 3 3 ) 2 = 5 3S = 35 = 243F 2.7 ก Fn ˈ ก F กn F ก ก 7 + 15 + 23 + n n+1 2n F F ก 217 F 2 + 2 + ... + 2 8 F Fก F 2 ก ก 7 + 15 + 23 + Fก F Sn = 217 ก ก Sn = n [ 2a1 + (n − 1)d ] 2 F a1 = 7, d = 8 F F 31
  35. 35. FF 217 = n [ 2(7)+ (n − 1)8 ] 2 = n ( 6 + 8n ) 2 กF ก FF n=7 (F F F) ก 2n + 2n + 1 + + 22n = 27 + 28 + + 214 a1 (1 − r n ) ˈ ก ก ก กn F ก F ก Sn = 1 − r F n = 14 7 + 1 = 8, a1 = 27, r = 2 FF 7 − 8 S7 = 2 (1− 22 ) 1 − = 128(1−1 256) = 128(−255) 1 2 n + 2 n+1 + ... + 2 2n = 128(255) = 127.5 28 256 ʿก 2.2 1 n = 1, 21. ก F an = an 2 + 2 n = 3, 5, 7 2an 2 n = 4, 6, 8, 101 F ∑ ai i=12. ก ก ก 100 F ก 5 ก 2 ʾ ก F3. ก {100, 101, 102, , 600} F 8 12 Fก F32 ก

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