高等数学英文1-4-10

lim xn = A ⇔
n→∞
∀ε > 0, ∃N > 0, s.t. ∀n > N ⇒ xn − A < ε
§1.4 The Limits of Sequences
Exercises of the limit
Prove the following limits.
sin x 1. lim =0 x → +∞ x
ε − N Def :
lim xn = L ⇔ ε > 0, ∃N > 0, s.t. ∀n > N ⇒ xn − L < ε ∀
n→∞
§1.4 The Limits of Sequences
II. Limit of Infinite Sequence
lim xn = a ⇔ ε > 0, ∃N > 0, s.t. ∀n > N ⇒ xn − a < ε ∀
x→a
then ∃δ > 0, when x ∈U(a,δ ), f ( x) > 0(or f ( x) < 0).
o
Corollary: Supposethat lim f ( x) = A, then ∃δ > 0,
x→a
A when x ∈U(a,δ ), f ( x) > 2
0
Th: Supposethat lim f ( x) = A, and∃δ > 0, when x ∈U(a,δ ),
n−1 Proof Let ε be given. If we want q − 0 = q n−1
<ε ,
that is
n > 1+ log q ε .
We choose N = 1+ log q ε , + Then n> N implies that q > That is limqn−1 = 0 .
x1, x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10,L x2 , x3 , x5 , x8 , x10,L
xn1 , xn2 , xn3 , xn4 , xn5 ,L
xn1 , xn2 ,L, xnk ,L
nk ≥ k
Theorem Any subsequence converges to the same number with the original sequence. Th: If x2k →a, x2k+1 →a(k →∞), then xn →a(k →∞).
∃M > 0, f ( x) ≤ M.
§1.4 The Limits of Sequences
III. Properties of the Limit
3. Conservation of the sign Th: Suppose that lim f ( x) = A, and A > 0(orA < 0),
n→∞
Geometric interpretation
a −ε
x 2 x1 x N + 1
2ε
a+ε
x
a x4 x N + 2 x 3
When n> N, all the points xn are in the neighborhood > U(a,ε ) where there are finite points (at most N points) not in .
• •••
1
3 2 5 4 0, , , , , L 2 3 4 5
3 2 5 4 0, , − , , − , L 2 3 4 5
1 an = 1+ (−1) n
an = (−1)n + 1 n
ห้องสมุดไป่ตู้
-1
•• -1
•• • • 1 • •
• 0
1
• 1
0.99, 0.99, 0.99, 0.99, L an = 0.99
3n + 1 3 2. lim = n → ∞ 2n + 1 2
1 1 3. lim = x→a x a
§1.4 The Limits of Sequences
1. Uniqueness Th: Let lim f ( x) = A, lim f ( x) = B, then A = B.
x→a x→a
2. Local boundedness Th: Let lim f ( x) = A, then ∃δ > 0, when x ∈U(a,δ ),
x→a 0
an = 3n − 2
a1 = 1, an = an−1 + 3, n ≥ 2
§1.4 The Limits of Sequences
I. Infinite Sequence
For instance:
1 2 3 4 0, , , , , L 2 3 4 5
an = 1− −
1 n
n
-1
• 0
• 0
§1.4 The Limits of Sequences
1+ n −1 < ε n
II. Limit of Infinite Sequence
Def: The sequence { xn} is said to converge to L, and we write
lim xn = L
n→∞
if for each given number
1. Uniqueness Th: Let xn →a, xn →b(n →∞), then a = b. 2. Boundedness Th: Let xn →a(n →∞), then xn is bounded .
§1.4 The Limits of Sequences
III. Properties of the Limit
Chapter 1 Functions and Limits
§1.4 The Limits of Sequences
I. Infinite Sequence
a1, a2 ,L, an ,L
is an ordered arrangement of real numbers. Formal Definition Infinite sequence is a function whose domain is the set of positive integers and whose range is a set of real numbers. explicit formula recursion formula
§1.4 The Limits of Sequences
Review of the limit
lim f ( x) = A ⇔
x→a
∀ε > 0, ∃δ > 0 , s.t 0 < x − a < δ ⇒ f ( x ) − A < ε
lim f ( x) = A ⇔
x→∞
∀ε > 0, ∃M > 0 , s.t x > M ⇒ f ( x ) − A < ε
ε , there is a corresponding
positive number N such that n > N ⇒ xn − L < ε . A sequence that fails to converge to any finite number L is said to diverge, or to be divergent.
-1
0
§1.4 The Limits of Sequences
Q: Do they converge to 1?
II. Limit of Infinite Sequence
Relationship with the two limits
1+ x f ( x) = x
1+ n f (n) = n
∀ε > 0,
§1.4 The Limits of Sequences
II. Limit of Infinite Sequence
Example 1
1. Prove that the lim of the sequence 1, q, q2 ,L, qn−1,L it is 0, provided that q < 1.
n→∞
[
]
n−1
<ε
§1.4 The Limits of Sequences
II. Limit of Infinite Sequence
Example 1
2. Prove that lim ( n + 1 − n ) = 0
n→ ∞
§1.4 The Limits of Sequences
III. Properties of the Limit
x→a
0
f ( x) ≥ 0(or f ( x) ≤ 0), then A ≥ 0(or A ≤ 0).
§1.4 The Limits of Sequences
III. Properties of the Limit
3. The convergence of subsequence
x1, x2 ,L, xi ,Lxn ,L