The reason I started this website was to post about the things I learned about each week on KA. This week I feel like I didn’t learn anything so I don’t really know what I’m going to talk about here. It wasn’t due to a lack of effort. I probably spent close to 10 hours working on the single exercise from the section Lagrange Error Bound, going through the 4 videos from that section (for the 3^rd or 4^th time) and also Googling and watching other videos on Taylor Series that I thought might help me understand what was going on. After all of that, I don’t I’m any closer to understanding the concept. I’m very frustrated, to say the least, and had a moment when I genuinely thought maybe I reached a point where I just wouldn’t be able to go any further. I eventually decided that if I can’t figure it out, worst case scenario, I’ll just go back and rewatch all the video in this unit leading up to this section to get a better grasp on Taylor Series, in general. Since my understanding of what I went through before this is tenuous at best, maybe that’s why I don’t understand what’s going on in this section.

Below are 4 examples of questions I worked through and the technique I tried to use to figure them out. Before I go through the examples, I’ll first explain how I tried to answer the questions which sometimes worked but sometimes didn’t. The way each question worked was it asked me to find the ‘error’, i.e. the difference from a Taylor Polynomial and the actual function in question, at a given value. As you’ll see from the KA answers, there’s a formula to do this which is:

• |R_n(x)| ≤ | M(x – x)^{n + 1}/(n + 1)!|

In layman’s terms, this states that the difference between the T.P. at x and the function at x will be less-than-or-equal-to some max Value, M. You can then use this formula to figure out what n (a.k.a. what derivative) you need to go to in order to find an error/remainder that’s less-than a given value (which I’m pretty sure was 0.001 in every question.). From watching some of the other videos I found on YouTube, I’m pretty sure M is:

• M = f^{(n + 1)}(z)

– where f^{(n + 1)} is the following derivative from the T.P. where the error is reached and z is a number between x and c that gives you the greatest M. (Also, x is the value in the function being approximated and c is the center of the polynomial.)

To be completely honest, I don’t really know what I’m talking about which is exactly why I didn’t make it through the exercise this week. I couldn’t understand how to use the formula so I tried to solve the questions by writing out what I thought was the T.P. into Desmos and kept adding degrees to the T.P. until the function and the T.P. at the x-value in question was less-than the error the question asked me for. Sometimes this worked but sometimes it didn’t and I couldn’t understand why.

So, with all of that, here are the 4 questions I worked on:

Question 1

As you can see from this question, my technique happened to work here as I clearly went to the 6^th degree T.P. before stopping.

Question 2

I worked through this question on Thursday and can’t remember why I got it wrong. Looking as my work on Desmos, clearly the 5^th degree was the first T.P. that had an error less-than 0.001 and not the 4^th degree… So I have no idea why I didn’t answer it as 5. Also, I don’t remember why in my notes I went all the way to the 6^th degree. I may have just done that so I had the equation ready to input into Desmos. So, even though I got this question wrong, I think my technique actually did work for this question… 🤔

Question 3

Not much to say about this question. My technique worked. Boom.

Question 4

This is an example where my technique didn’t work. I wrote out the T.P. into Desmos and when I reached the 8^th degree and subtracted |f(1.6) d – p₈(1.6)| the output was ~0.0007 which was less-than the error I was asked to find, 0.001. The answer apparently was 9 though which doesn’t make sense to me. I can’t really follow along with the algebra from KA’s answer using R(x) so I think I must be missing something here…

Two things I’m going to do this coming week are 1) try to get a better understanding of the R_n(x) formula and 2) also keep Googling other videos on T.P.’s to try and figure out what’s going on more broadly. I found a pretty good playlist on YouTube that covers T-Series and T-Polynomials so I’m going to keep watching that playlist. There’s no chance I’ll end up getting through this unit, Series (1,120/2,000 M.P.), before the start of the New Year, and it’s a bit of a relief to acknowledge that. I don’t really care that it’s not going to happen. I’m more concerned with going back and figuring out what’s actually going on with T-Series, in general. I’m sure I’ll figure it all out eventually as long as I keep working away. Two steps forward and one step back I guess. ➡️➡️⬅️