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1
What is a sequence?
2
Limit of a Sequence
3
The Basics (Limit Example 1)
4
Simple Fraction (Limit Example 2)
5
DON'T factor out! (Limit Example 3: Complex Fraction)
6
The Limit Does NOT Exist (Limit Example 4)
7
Limits are unique
8
Square Root Limit (Limit Example 5)
9
Bounded Away from Zero
10
Sum of limits
11
Convergent sequences are bounded
12
Absolute Value (Limit Example 6)
13
Squeeze Theorem Proof
14
Product of Limits
15
Reciprocals (Limit Example 7)
16
Exponentials (Limit Example 8)
17
DON'T use logarithms (Limit Example 9)
18
Infinite Limits (Limit Example 10)
19
Infinite Limit Laws
20
Limit Duality Theorem
21
Power vs Exponential limit (Limit Example 11)
22
Monotone Sequence Theorem
23
Golden limit
24
Babylonian Square Root
25
Monotone Sequence implies Least Upper Bound
26
Decimal Expansions
27
What is Limsup ?
28
Limsup vs Limits
29
Limsup Squeeze Theorem
30
Cauchy Sequences
31
Completeness
32
The Legend of Z
33
Real numbers Cauchy construction
34
What is a Subsequence?
35
Inductive Construction
36
Another Inductive Construction
37
Monotone subsequence
38
Can limsup be attained?
39
Limit Points
40
The Bolzano Weierstraß Theorem
41
Direct Bolzano Weierstraß
42
Can you solve this UC Berkeley math PhD entrance exam question?
43
A neat diagonal argument
44
Limsup Product Rule
45
Limsup Sum Rule
46
Pre-Ratio Test
47
Limit with n!
48
Not your average YouTube video
Description:
Explore the fundamental concepts of sequences in mathematics through a comprehensive 10-hour course. Delve into topics such as sequence limits, convergence, bounded sequences, and the Squeeze Theorem. Master complex limit examples, including fractions, square roots, and exponentials. Investigate advanced concepts like the Monotone Sequence Theorem, Limsup, Cauchy sequences, and the Bolzano Weierstraß Theorem. Tackle challenging problems, including a UC Berkeley math PhD entrance exam question, and discover unique mathematical arguments. Gain a deep understanding of sequence properties, limit laws, and their applications in mathematical analysis.

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#Mathematics #Calculus #Convergence #Real Numbers #Mathematical Analysis #Real Analysis #Cauchy Sequences
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What is a sequence?