Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE

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Published on Jul 09, 2020

About :

In this video, I go over an extremely extensive tutorial on Sequences as part of my new video series on Sequences and Series. Sequences are ordered lists of numbers where each term is categorized by its value and the integer corresponding to its place or order in the list. Series are summations of sequences; which I have shown in my earlier videos in finding areas and hence integrals of functions.

The topics I cover in this video include: Sequences and Notation, Definition and Precise Definition, Infinite Limits, Monotonic and Bounded Sequences, as well as the famous Fibonacci Sequence and its relationship with the also very famous Golden Ratio. And at the end of this video I go over a brief mainstream history overview of Leonardo Fibonacci. The Fibonacci sequence is the sequence in which the initial 2 terms are 1 and 1 (or 0 and 1 in modern computing) and then the terms that follow are just the summation of the previous two terms. The limit of the ratio of consecutive Fibonacci terms approach the Golden Ratio (φ) (which is the ratio of any two numbers such that it is the same as the ratio of their summation to the larger number). Fibonacci and Golden numbers/sequences/ratios/spirals/angles/etc. appear very often in all aspect of life from art, nature, design, finance, and even advertising! #AmazingStuff

Here is a list of the topics covered and their start times in the video:

@ 55:54 - Introduction to Sequences and Series
@ 9:25 – Sequences and Notation
@ 11:06 - Examples 1 to 3
@ 27:30 – Limit of a Sequence
@ 33:59 – Definition 1: Limit of a Sequence
@ 37:38 – Definition 2: Precise Definition of a Sequence
@ 47:33 – Theorem 1
@ 52:26 – Definition 3: Infinite Limits of Sequences at Infinity
@ 58:17 – Limit Laws for Sequences
@ 1:01:39 – Squeeze Theorem for Sequences
@ 1:08:10 – Theorem 2
@ 1:08:51 - Examples 4 to 7
@ 1:18:51 – Theorem 3
@ 1:19:48 – Examples 8 to 9
@ 1:26:46 – Creating Graphs of Sequences
@ 1:29:33 – Example 10
@ 1:41:46 – Definition 4: Monotonic Sequences
@ 1:43:42 – Examples 11 and 12
@ 1:54:08 – Definition 5: Bounded Sequences
@ 1:59:31 – Completeness Axiom
@ 2:03:43 - Monotonic Sequence Theorem
@ 2:09:27 – Example 13: Recursion and Mathematical Induction
@ 2:22:27 – Exercise 1: Convergence
@ 2:33:20 - Exercise 2: Proof of Theorem 2
@ 2:39:41 - Exercise 3: Proof of Theorem 3
@ 2:48:23 - Exercise 4: Fibonacci Sequence
@ 3:04:17 – Mainstream History of Fibonacci
@ 3:12:04 – Overview of Fibonacci Numbers
@ 3:17:33 – Overview of the Golden Ratio
@ 3:21:53 – Relationship Between Fibonacci Sequence and Golden Ratio
@ 3:27:06 – Golden Angle/Spiral and Approximations with Fibonacci Spiral
@ 3:31:30 – Fibonacci Numbers and Golden Ratio in Nature, Art, Design, & MORE! #Amazing