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

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    mes

    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:

    1. @ 55:54 - Introduction to Sequences and Series
    2. @ 9:25 – Sequences and Notation
    3. @ 11:06 - Examples 1 to 3
    4. @ 27:30 – Limit of a Sequence
    5. @ 33:59 – Definition 1: Limit of a Sequence
    6. @ 37:38 – Definition 2: Precise Definition of a Sequence
    7. @ 47:33 – Theorem 1
    8. @ 52:26 – Definition 3: Infinite Limits of Sequences at Infinity
    9. @ 58:17 – Limit Laws for Sequences
    10. @ 1:01:39 – Squeeze Theorem for Sequences
    11. @ 1:08:10 – Theorem 2
    12. @ 1:08:51 - Examples 4 to 7
    13. @ 1:18:51 – Theorem 3
    14. @ 1:19:48 – Examples 8 to 9
    15. @ 1:26:46 – Creating Graphs of Sequences
    16. @ 1:29:33 – Example 10
    17. @ 1:41:46 – Definition 4: Monotonic Sequences
    18. @ 1:43:42 – Examples 11 and 12
    19. @ 1:54:08 – Definition 5: Bounded Sequences
    20. @ 1:59:31 – Completeness Axiom
    21. @ 2:03:43 - Monotonic Sequence Theorem
    22. @ 2:09:27 – Example 13: Recursion and Mathematical Induction
    23. @ 2:22:27 – Exercise 1: Convergence
    24. @ 2:33:20 - Exercise 2: Proof of Theorem 2
    25. @ 2:39:41 - Exercise 3: Proof of Theorem 3
    26. @ 2:48:23 - Exercise 4: Fibonacci Sequence
    27. @ 3:04:17 – Mainstream History of Fibonacci
    28. @ 3:12:04 – Overview of Fibonacci Numbers
    29. @ 3:17:33 – Overview of the Golden Ratio
    30. @ 3:21:53 – Relationship Between Fibonacci Sequence and Golden Ratio
    31. @ 3:27:06 – Golden Angle/Spiral and Approximations with Fibonacci Spiral
    32. @ 3:31:30 – Fibonacci Numbers and Golden Ratio in Nature, Art, Design, & MORE! #Amazing

    Stay tuned for my next super long math video!

    Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh68vMw9e1A75d63O3w

    View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more

    Related Videos:

    Limits: https://www.youtube.com/playlist?list=PLFE39AF0D7E0651AC
    Areas and Summation: https://www.youtube.com/playlist?list=PLai3U8-WIK0E6biFIz0glhAnUwE1qaviI
    Parametric Equations and Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0H0AMIZV8HU3LVxSSf_Y9pi .


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