The Precise Definition of a Limit of a Sequence

In this video I go over a more precise definition of the limit of a sequence by using the same approach as for the limit of a function. If the limit of a sequence exists, then there must be a positive number ε such that it has a corresponding number N where if the sequence term n is greater than N, then the absolute value of the difference between the n-th term of the sequence and the limit L is less than ε. Basically this means that even if we approach an infinite number of terms, the number ε just defines a smaller and smaller interval by which the difference between the terms of the sequence and the limit must be within. I also show that the precise definition of the limit of a function is the same as that for the sequence if the limit of the function applies for positive integers. I illustrate this with an example as well.

#maths #science #sequences #algebra #education

Timestamps:

- Definition 2: Precise definition of the limit of a sequence: 0:00
- Plotting the definition on a number line: 1:47
- Graphing the precise definition: 5:32
- Recap on the precise definition of a the limit of a function: 9:02
- N is any real number for functions, and only integers for sequences.
- Theorem 1: Limit of a function is the same as the limit of sequence if n can be an integer: 9:54
- Graphing out Theorem 1 shows the sequence on the function: 10:56
- Example: Limit of a function is zero, thus its corresponding sequence also approaches zero: 12:18

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-2ly9qbk9k
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUgfK34Voi6BvrNqt4X4BV
- Notes: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
- Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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