This week was an interesting one. After 3 weeks I FINALLY made it through the exercise in the section Lagrange Error Bound and, believe it or not, am pretty sure I now understand what I was doing wrong. I then crushed the following section, Power Series Intro, and got through the two vids and 1 exercise in ~2 hours. Finally, I watched the first 4 videos in the next section, Function as a Geometric Series, but unfortunately I don’t have a great understanding of what’s going on in that section. The reason I say this week was interesting was because although I definitely got a better understanding of series in general, I realized there’s WAY more to these types of series than I thought there was. I find this happens to me often where the more I learn about a certain topic, the more I realize it goes much deeper than I first thought. Nonetheless, I 100% making progress understanding everything which is a relief especially considering I was stuck on the same exercise for the past 3 weeks.

To get straight to the point, here’s a question from the Lagrange Error Bound exercise and a page from my notes that sums up what I think I was doing wrong:

For the past few weeks I’ve been inputting the T.P.’s into Desmos which would tell me what the first derivative ,n, was where the T.P.’s error was less-than 0.001, which was always the error each question asked for. As far as I understand, I’m pretty sure that’s not what the exercise wanted me to do. Even though my evaluation of the T.P. using Desmos DID give me the ‘correct’ and precise n where the T.P.’s error was less-than 0.001, I think the whole point of this exercise was to use the Lagrange Error Bound theorem to come up with an error less-than 0.001 because sometimes you don’t have a calculator (I guess?) and need to know for sure that the n you use will give you an error less-than whatever. I don’t actually know if I’m right about all of this but that’s what it seems like.

As you can see from the second screen shot above, in the 3^rd row you can see that the T.P., p(x), went to the third-degree (the second-degree is missing because it equals 0 since f’’cos(1.5π) = -cos(1.5π) = 0) and in the 5^th row you can see the absolute value of |f(1.3π) – p(1.3π)| = 0.000808423814914 < 0.001. This means that in reality the third-degree T-polynomial of cos(x) gives you an error less-than 0.001, BUT, as you can see in the 6^th row of the third screen shot, when you use the third-degree in the Lagrange Error Bound theorem the output is ~0.04 > 0.001. You have to go to the fifth-degree before the error gets to less-than 0.001 and since the theorem uses n + 1 and the degree you need to go to is 5, that means the minimum n is 4. (I just read that over as I’m editing this post and would be amazed if anyone actually understood what I just wrote.)

Using this approach meant that I essentially needed to memorise the formula M(x – c)ⁿ⁺¹/(n + 1)! where M = fⁿ⁺¹(z) and z was whatever value on the interval from x to c that produced the largest value for M. Most often z was either x or c (except for when they gave me a trig T.P. where z was 1 every time). Once I had a better grasp on the formula, instead of writing out the entire T.P. into Desmos I just started inputting the values for the theorem into Desmos. Here’s an example:

Once I started to do this, the questions became much easier to understand and get to the correct solution. The tricky part to remember was that when I finally got to an n that resulted in the error being less-than 0.001, the correct n to use was actually the n before since the theorem uses n + 1.

I started the next section, Power Series Intro, on Thursday and, as I mentioned at the beginning, got through it in just a few hours. I figured out how to work through the questions from the exercise but I don’t really know why they work the way they do. Here’s a question from the exercise:

To be honest, I don’t really understand how series represent functions, in general, and how they sometimes only represent certain functions across a specific interval. My understanding is that series sometimes represent functions where the interval is from negative to positive infinity but a lot of the time the interval is only across a small interval like from –1 < x < 1 or something like that.

The thing I’m starting to get a better grasp on is that n isn’t really a variable in the way that I’ve typically thought about it. In these questions, n implies that the given formula is a bunch of terms that each have n in them, that are being added together, and where the n in each term monotonically increases (i.e. the n goes up by 1 in each successive term). I’m having a hard time putting this concept into words because I don’t really understand it that well, BUT I can tell that it’s starting to become more clear to me.

Finally, I started the section Function as a Geometric Series on Friday and got through the first 4 videos but don’t really understand what’s going on at all in that section so I’m not going to try and explain it this week. I only have 1 video and 1 exercise left to finish off the section so I’m hoping I’ll get through it early next week and then I’ll explain it in my next post.

It’s officially 2023 and I definitely did not finish this unit Series (1,280/2,000 M.P.) let alone the entire subject Calculus 2 (9,780/10,500 M.P.), but I’m fine with that. Over the past few months, I’ve had to come to terms with the fact that calc is frickin hard and that it’s taking me a lot longer to learn than any other subject I’ve worked through up to this point. Even though I’ve come to terms with that, it’s definitely frustrating at times but it makes me think of a video I saw a few months ago of Neil DeGrasse Tyson where he talked about the jump from algebra to calculus. Remembering what he says in this video helps me to not feel quite so bad about how long it’s taking me. I’m 100% confident I’m going to get there eventually, I just need to keep grinding away. Here’s the vid: