Whoops, sorry about the mistakes on that “Raw Form” slide at 23:03. I provided the corrections below, as well as 2 things that I didn't explain in the video that I should clarify: Correcting Mistakes at 23:03 (1): I messed up pretty much the ENTIRE procedure for substitution of "n" for "i" in that one "Raw Form" slide at 23:03 - It's a wild mess!! I still got the right answer, but I provided the wrong explanation. Let's start with the definition of the variable "n". According to that one slide, the correct equation for n SHOULD BE: n = i + 2 (and also N = I + 2). Thus, the first term in the sequence should ACTUALLY correspond to n = 0, NOT n = 1. And the second mistake I made was my explanation for WHY we can simplify to 1/n! . It SHOULD BE because of the fact that the "repeated multiplication of 2 + j from j = 0 to j = i" is equivalent to "repeated multiplication of 2 + j from j = 0 to n -2", and this is equal to "n!" . And we can see this if we say that k = j + 2: We have n! = "repeated multiplication of k from k = 2 to k = n". And this is equal to n*(n - 1)*(n - 2)*(n - 3)*...*2 . And, if we want, we can multiply by 1 also, so this is indeed equal to n! Clarification 1: I don’t think I explained well enough WHY the two forms of what I called the "expandable integral terms" (you know, for q = + or - 1 AND m = even or odd) are indeed the ONLY 2 forms of the expandable terms you will encounter in the Taylor Series. This is highlighted by starting with your general form, expanding it out to 2 more terms. You will then notice that if you DON’T simplify using the distributive property, then you obtain an expandable term at the end that is of the form of the ORIGINAL expandable term from 2 terms back in the sequence. And I show this in the presentation, but I do not HIGHLIGHT this important fact. Clarification 2: I saw online that people express the "error term" as a term that does not involve an integral. So, that surprised me, and maybe I'll try to figure out how to simplify that integral and make a video about it in the future.
My own opinion of this video: the content and script were fantastic, but the audio can be improved: I think that the headphones made my voice a bit too muffled, and I got a little too “breathy” and “tongue-clicky”. And I unnecessarily put emphasis on words too many times. I’ll try to fix these problems with audio and narration in future videos
Whoops, sorry about the mistakes on that “Raw Form” slide at 23:03. I provided the corrections below, as well as 2 things that I didn't explain in the video that I should clarify:
Correcting Mistakes at 23:03 (1):
I messed up pretty much the ENTIRE procedure for substitution of "n" for "i" in that one "Raw Form" slide at 23:03 - It's a wild mess!! I still got the right answer, but I provided the wrong explanation. Let's start with the definition of the variable "n". According to that one slide, the correct equation for n SHOULD BE: n = i + 2 (and also N = I + 2). Thus, the first term in the sequence should ACTUALLY correspond to n = 0, NOT n = 1. And the second mistake I made was my explanation for WHY we can simplify to 1/n! . It SHOULD BE because of the fact that the "repeated multiplication of 2 + j from j = 0 to j = i" is equivalent to "repeated multiplication of 2 + j from j = 0 to n -2", and this is equal to "n!" . And we can see this if we say that k = j + 2: We have n! = "repeated multiplication of k from k = 2 to k = n". And this is equal to n*(n - 1)*(n - 2)*(n - 3)*...*2 . And, if we want, we can multiply by 1 also, so this is indeed equal to n!
Clarification 1:
I don’t think I explained well enough WHY the two forms of what I called the "expandable integral terms" (you know, for q = + or - 1 AND m = even or odd) are indeed the ONLY 2 forms of the expandable terms you will encounter in the Taylor Series. This is highlighted by starting with your general form, expanding it out to 2 more terms. You will then notice that if you DON’T simplify using the distributive property, then you obtain an expandable term at the end that is of the form of the ORIGINAL expandable term from 2 terms back in the sequence. And I show this in the presentation, but I do not HIGHLIGHT this important fact.
Clarification 2:
I saw online that people express the "error term" as a term that does not involve an integral. So, that surprised me, and maybe I'll try to figure out how to simplify that integral and make a video about it in the future.
My own opinion of this video: the content and script were fantastic, but the audio can be improved: I think that the headphones made my voice a bit too muffled, and I got a little too “breathy” and “tongue-clicky”. And I unnecessarily put emphasis on words too many times. I’ll try to fix these problems with audio and narration in future videos