The Precise Definition of a Limit of a Sequence

mes (67)in DTube • 2 days ago

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: