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

3. 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 F
1.5 กFก F Fก F F ʿก F
F FF F Fก F FFF ก F F
F ก F ก
F
16 ก . . 2549

5. 1 1 24
1.1 1
1.2 3
1.3 5
1.4 7
1.5 กFก 20
1.6 ʾ ก 23
2 ก 25 36
2.1 ก 25
2.2 ก 29
2.3 ก 33
3 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

7. 1
1.1
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

8. 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 1.1
1. 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, } → S1
2 ก

9. 4) f2 : {1, 3, 5, } → R
5) g2 : {1, 2, 3, } → S2
6) h2 : S3 → R
2. 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)d
a1 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

10. 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 3n
4 ก

11. 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 1.2
1. 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 F
2. F ก ˈ 200, 182, 164, 146, F F ก F 10 F ก F
1.3
1.4
(geometric sequence) F F F ก Fก
F ก F F F (common ratio)
ก 1.4 FF
F F 5

12. 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 1.3
1. 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 F
2. ก F a + 3, a, a 2 ˈ F ก F F ˈ r F F
∞ n −1
∑ ar Fก F
n=1
3. F x, y, z, w ˈ F4 F ก x ˈ F ก F y+z = 6
z + w = 12 F F F 5
6 ก

13. 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 = 1
n→∞
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 L
n→∞
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

14. 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 ก
8 ก

15. 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 ก
F 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

16. F ˈ ก ก F ก ก F กF ˆ
ก 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 2
10 ก

17. lim ( a n + b n ) = L + M
n→∞
4) an bn = an + ( bn) ก F 3) ก F F ก
5) 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→∞
an
6) b n = an ⋅ b n bn ≠ 0 ก F 5) ก F F ก
1
7) 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

18. 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′ n
12 ก

19.    
=  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
1
2) 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 nk
4) 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
1
5) 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

20.  
= 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
3
14 ก

21. 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
= 2
3) lim ( a n + b n ) = lim a n + lim b n
n→∞ n→∞ n→∞
= 2+23
3
= 13
6
4) lim ( a n ⋅ b n ) = lim a n ⋅ lim b n
n→∞ n→∞ n→∞
= 2 ⋅2
3
3
= 1
lim a n
5) lim
n→∞
( ) an
bn = n →∞
lim b n
n →∞
2
= 3
3
2
= 4
9
F F 15

22. 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

23. 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 F
ก 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

24. 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→∞ 2
18 ก

25. ʿก 1.4
1. 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 n
5. 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

26. 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 กFก
20 ก

27. 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 F
F 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

28. ʿก 1.5
F ก FF F F F ก F F F
1. an = ( −1) n nn
2
2
2. an = ( −1) n n n
2
−n
3. an = ( −1) n e 3
n
n sin n
4. an = ( −1) 2
n
n n
5. an = ( −1) ln n
22 ก

29. 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

30. 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 1.6
1. F n ก F F ʾ ก
1) n = 9
2) n = 13
3) n = 16
2. F ก กก F 100 ʾ ก ก
24 ก

31. 2
ก
2.1 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

32. 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
4
26 ก

33. 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
2047
F 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

34. ʿก 2.1
1. 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 2
3. F F F ∑ ( ai + bi ) ≤ ∑ ai + ∑ bi
2
ก กn
i=1 i=1 i=1
10 10 10 10
4. 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=1
5. 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 =1
6. F sin21° + sin22° sin23° + sin289°
28 ก

35. 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 2.3
ก 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

36. = 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
2
30 ก

37. 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 = 243
F 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

38. 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, 2
1. ก F an = an 2 + 2 n = 3, 5, 7
2an 2 n = 4, 6, 8,
101
F ∑ ai
i=1
2. ก ก ก 100 F ก 5
ก 2 ʾ ก F
3. ก {100, 101, 102, , 600} F 8 12 Fก F
32 ก