Advanced Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
WEN-CHINGLIEN Advanced Calculus (I)
2.1 Limits Of Sequences
Definition
A sequence of real numbers {xn} is said to converge to a real number a ∈R if and only if for every > 0 there is an N ∈N (which in general depends on ) such that
n ≥ N implies |xn− a| <
WEN-CHINGLIEN Advanced Calculus (I)
2.1 Limits Of Sequences
Definition
A sequence of real numbers {xn} is said to converge to a real number a ∈R if and only if for every > 0 there is an N ∈N (which in general depends on ) such that
n ≥ N implies |xn− a| <
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Example:
Prove that 1
n → 0 as n → ∞.
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Example:
Prove that 1
n → 0 as n → ∞.
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Example:
The sequence {(−1)n}n∈N has no limit.
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Example:
The sequence {(−1)n}n∈N has no limit.
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Remark:
A sequence can have at most one limit.
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Remark:
A sequence can have at most one limit.
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Definition
By a subsequence of a sequence {xn}n∈N, we shall mean a sequence of the form {xnk}k ∈N, where each nk ∈ N and n1 <n2 < . . .
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Definition
By a subsequence of a sequence {xn}n∈N, we shall mean a sequence of the form {xnk}k ∈N, where each nk ∈ N and n1 <n2 < . . .
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Remark:
If {xn}n∈N converges to a and {xnk}k ∈N is any
subsequence of {xn}n∈N, then xnk converges to a as k → ∞.
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Remark:
If {xn}n∈N converges to a and {xnk}k ∈N is any
subsequence of {xn}n∈N, then xnk converges to a as k → ∞.
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Definition
Let {xn} be a sequence of real numbers.
(i){xn} is said to be bounded above if and only if there is an M ∈R such that xn ≤ M for all n ∈ N
(ii) {xn} is said to be bounded below if and only if there is an m ∈R such that xn ≥ m for all n ∈ N
(iii) {xn} is said to be bounded if and only if it is bounded both above and below.
WEN-CHINGLIEN Advanced Calculus (I)
Definition
Let {xn} be a sequence of real numbers.
(i){xn} is said to be bounded above if and only if there is an M ∈R such that xn ≤ M for all n ∈ N
(ii) {xn} is said to be bounded below if and only if there is an m ∈R such that xn ≥ m for all n ∈ N
(iii) {xn} is said to be bounded if and only if it is bounded both above and below.
WEN-CHINGLIEN Advanced Calculus (I)
Definition
Let {xn} be a sequence of real numbers.
(i){xn} is said to be bounded above if and only if there is an M ∈R such that xn ≤ M for all n ∈ N
(ii) {xn} is said to be bounded below if and only if there is an m ∈R such that xn ≥ m for all n ∈ N
(iii) {xn} is said to be bounded if and only if it is bounded both above and below.
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Definition
Let {xn} be a sequence of real numbers.
(i){xn} is said to be bounded above if and only if there is an M ∈R such that xn ≤ M for all n ∈ N
(ii) {xn} is said to be bounded below if and only if there is an m ∈R such that xn ≥ m for all n ∈ N
(iii) {xn} is said to be bounded if and only if it is bounded both above and below.
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Definition
Let {xn} be a sequence of real numbers.
(i){xn} is said to be bounded above if and only if there is an M ∈R such that xn ≤ M for all n ∈ N
(ii) {xn} is said to be bounded below if and only if there is an m ∈R such that xn ≥ m for all n ∈ N
(iii) {xn} is said to be bounded if and only if it is bounded both above and below.
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Theorem
Every convergent sequence is bounded.
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Theorem
Every convergent sequence is bounded.
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Exercise:
1 lim
n→∞
5 + n n2
2 Suppose that lim
n→∞xn=1 Find lim
n→∞
2 + xn2
xn
WEN-CHINGLIEN Advanced Calculus (I)
Exercise:
1 lim
n→∞
5 + n n2
2 Suppose that lim
n→∞xn=1 Find lim
n→∞
2 + xn2
xn
WEN-CHINGLIEN Advanced Calculus (I)
Exercise:
1 lim
n→∞
5 + n n2
2 Suppose that lim
n→∞xn=1 Find lim
n→∞
2 + xn2
xn
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Thank you.
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