This week was another one where I felt like I was finally starting to get a solid grasp on T-series and T-polynomials and yet at the same time felt just as confused as ever about them. 😔 I do think I’m starting to get a better grasp on how they work, i.e. memorizing the formulas and the notation, but I’m still getting stuck on certain questions mainly because I don’t understand why the formulas work… It’s a bit confusing and disappointing that sometimes I feel like I’m crushing this section and understand exactly what’s going on but then I come up against a question where I get rocked. Now that I think about it, Taylor series and Taylor polynomials have been one of the most demoralizing things I’ve worked through on KA. I know I’m going to get through it but I feel pretty defeated right now. 😰

BUT NONTHELESS!!!

I actually started the week off pretty well by getting through the one exercise from the section Function as Geometric Series on my first attempt. Although I got through it relatively easily on Tuesday, I only understood how the pattern of the questions worked but didn’t have a clue why they worked. Here are two questions from that exercise:

Question 1

Question 2

In order to figure out the solution to these types of questions, you need to know the formula a/(1 – r). As you can tell from the top of notes in the first question, I don’t really understand why –6x = r or how 1/(1 + 6x) = a/(1 – r) = _n=0Σ^∞ a_n(x)ⁿ. The pattern is pretty straightforward though as the numerator is simply a and then you keep multiplying r by itself n-number of times to get the pattern.

After I finished that exercise, I began the following section Maclaurin Series of e^x, sin(x), and cos(x) on Wednesday. This section helped me to retain the formula for Maclaurin T.P.’s which was very useful. There were 8 videos in this section and the last video worked through what’s known as Euler’s Identity which I found pretty mind-blowing and which combines the Mac-polynomials of e^x, cos(x), and sin(x). Before I talk about Euler’s Identity (which is pronounced “Oil-er’s Identity”), here are my notes for how to derive the series for cos(x), (sin(x), and e^x:

When I was working through the section Lagrange Error Bound last week, there were questions in the exercise that used cos(x) and sin(x) which I had a hard time figuring out. Working through this section on the Mac-P’s for each trig function this week made those questions from last week much easier to understand, in retrospect. (Come to think of it, it seems like it would have been better for KA to have put this section, Maclaurin Series of e^x, sin(x), and cos(x), before the section Lagrange Error Bound.)

Two notable things about the cos(x) and sin(x) Mac-P’s are that 1) they each alternate and 2) that cos(x) has terms which go to even powers, (i.e. 2, 4, 6, etc.) and that sin(x) has terms that go to odd powers (i.e. 1, 3, 5, etc.). Once you figure out the Mac-P’s for cos(x), sin(x), and e^x, you can then combine them using algebra and with i, the imaginary number which equals the square-root of –1, and with π to get Euler’s Identity:

Sal points out in the video on Euler’s Identity that it’s pretty insane considering that the numbers e, i, π, 1, and 0 are all highly significant numbers in math and they’re all somehow combined/involved with this single identity with no other numbers included. Seeing and somewhat understanding this identity (at least somewhat understanding the how of this identity) is pretty mind-blowing for me and makes me think there are some bigger things happening out there on a cosmic scale that people don’t really understand. It makes me think there’s some god of math out there that’s pulling the strings of the universe.

I finished this week by working through the exercise from this section. I thought I had it all figured out but ended up getting the last question on the exercise wrong on my first and only attempt. Here are three questions from the exercise, the first two of which I got correct and the third one being the one I got wrong:

Question 3

Question 4

Question 5

I’m not going to try to explain what’s going on here (since I don’t actually know…), but part of what I needed to do to answer these questions was write out the first few terms of each series and identify which Mac-P between cos(x), sin(x), and e^x looked similar to the first few terms. To be completely honest, I don’t really know how I went about finding the solution to the questions once I did that. I kind of guessed what to do and managed to get the first 3 questions of the exercise correct. The fourth question in the exercise, which was the last question posted just above, was very confusing to me and one that I still don’t understand. I straight-up don’t understand why if x = ln(28) then the series converges to 28. I do understand why e^ln(28) = 28 but don’t see why that series would converge to 28. I’m hoping I’ll have it figured out by next week and can explain then.

Although I’m feeling a bit deflated, I’m still confident I’ll eventually get to the other side of Series (1,410/2,000 M.P.). I’ve said this many times before, but what I’m going through right now brings me back to the unit circle in Trig. I struggled with understanding that concept for at least a few weeks and probably didn’t really fully understand it until a few months had gone by of working on it. In retrospect the unit circle seems 100% clear and obvious to me now as to why it works. I’m sure that T-series and T-polynomials will eventually be just as clear to me but it’s going to take a bit more grit from me to get there. 😤