This week was the first time in what feels like forever that I’m actually happy with the amount of material I was able to get through. I only made it through 2 sections but that included 10 videos and 2 exercises. Everything I’ve worked on up until this week from the unit Series has all been review of how series work. This week I finally started to use series in calculus which was a relief in a weird sort of way. I think it’s partly because I enjoy doing calculus much more than what I’ve been reviewing on the math behind series. That said, I barely scraped the surface of ‘how’ Taylor Series work and literally have no understanding of ‘why’ they work or what they’re even talking about. That’s ok though. I’ve been in this situation before and will just keep grinding away at it until I eventually understand what’s going on.
My week started off working through the single exercise from the section Alternating Series Error Bound. I had attempted this exercise at the end of last week but wasn’t able to get through it and didn’t really understand what was happening. For the most, this week I was able to understand the math from this exercise. Here’s a question I worked through:
(I just realized I didn’t fill in the blanks of the KA answer before screen-shotting the question. 🤦🏻♂️ In order, the options were positive/negative and overestimate/underestimate.)
To understand what’s going on in this question you have to know that in an infinite, alternating series, the ‘error’ after any given term (i.e., the ‘remainder’ of the series taken after any term) is less-than the following term. In this question, the series is partially summed at n = 99, a.k.a. we’re finding the sum of the first 99 terms. The remainder of the series is therefore less-than 100th term. Inputting 100 into the formula gives you -0.01 meaning that “negative” is the answer to the first part of the question. Since the first term of the error is negative, that means that the sum of the first 99 terms is an overestimate of the actual sum of the infinite series since the reminder would reduce the value of the first 99 terms.
I got through that exercise on Tuesday and then began the following section, Taylor and Maclaurin Polynomials Intro, on Wednesday. This section had 6 videos and 1 exercise in it. I said this at the beginning of this post, but I have no clue what’s going on with Taylor or Maclaurin polynomials. (I do know, however, that a Maclaurin polynomial is a subset of Taylor polynomials that uses x = 0.) I’m certain that the following is at best a vague definition of Taylor polynomials and at worst completely wrong, but my understanding of a T.P. right now is that they’re a polynomial, p(x), (which is a type of series) based off a function, say f(x), where each successive term’s derivative is monotonically increasing (a.k.a. the second term is the first derivative of p(x), the third term is the second derivative of p(x), etc.).
I feel very confident that what I just wrote doesn’t make sense but, like I said, I literally don’t know understand I’m talking about right now… I DO know what the pattern for Taylor and Maclaurin series looks like, however:
To be clear, I don’t understand why these patterns work or what they’re even talking about. Here’s a screen shot from the third video of this section that works through a 2nd degree Maclaurin Polynomial:
I got though all 6 videos by and managed to finish the exercise by Thursday. I actually got through the exercise on my first attempt but I was a bit lucky and (for the millionth time) didn’t really understand what was going on. Nonetheless, here are 3 questions I worked through:
Question 1
Question 2
Question 3
I started the following section, Lagrange Error Bound, on Friday. I managed to get through all 4 videos in this section by Saturday but had even less of an understanding of what was happening with this concept than I did working through the previous section. I ended up going back to the earlier section and rewatching the first couple of videos to try and get a better grasp on everything. Here’s the one note I took from Lagrange Error Bound that just talks about the notation used to denote the error/remainder of a Taylor Series:
Hopefully I’ll have a better understanding of everything that’s going on next week so I can give a better explanation of Taylor series’ error and Taylor/Maclaurin series, in general.
I doubt I’m going to be able to get through Series (1,120/2,000 M.P.) by the end of December. I only have 6 exercises left but still have 25 videos left to get through. I’ll keep grinding away and hopefully I’ll be able to do it but I won’t be upset at all if it doesn’t happen, as long as I give it a decent effort. There were times this past week where I felt like I had no clue what was going on but there were also times when I crushed the algebra in the exercises and felt like an absolute genius. I’m 100% getting better at math even though the majority of the time I feel like I’m out of my depth. BUT, if there’s any lesson I’ve learned throughout all of this, it’s that you literally always feel out of your depth when you’re learning new concepts and that 1) it’s not the end of the world and that everything’s going to be fine, and 2) there’s no other way around it.