I finally made it to the unit test of Series this week but things didn’t go very well. I made it through 7 questions on the test and got 1 wrong. More of a disappointment than getting a question wrong was that of the 6 questions I got correct, I didn’t know what the hell I was doing in 5 of them and flat out cheated on 1 of them… 😔At the end of last week’s post I said that I’d probably have redo the unit test at least once. Working through the first 7 questions, I’m much further out of my depth than I thought I’d be (after 3 f@%king months of working on this) which is pretty demoralizing to say the least. The one potential bright spot from this week (which is a HUGE milestone if it’s actually true) is that I think I may have FINALLY figured out why T-series and T-polynomials follow along the shape of a function. I figured it out (I think) from a YouTube video which is at the bottom of this post. If this is true and I did in fact figure it out, it makes getting rocked by the Unit Test a lot easier to handle. In the big picture, understanding why T-series and T-polynomials work is more important than being able to demonstrate how to work through them without actually knowing what’s going on.

Before I started the unit test, I had to finish the final exercise from the section Representing Functions as Power Series. I finished this exercise on Tuesday and was able to get a decent grasp on how to work through the questions. The gist of how these questions worked was they’d give me either two functions, or series of terms, or series’ formulas which were similar to each other and I’d need to 1) recognize that one was the derivative/antiderivative of the other and 2) use derivatives/integrals to simplify one to determine the others’ formula/function. (I’m pretty sure that didn’t make sense but I don’t know how else to explain it…) Here are two example questions:

Question 1

Question 2

After working through that final exercise, I started the Unit Test on Wednesday. Of course all the questions had to do with working through different types of series, but they varied a fair amount in the specifics of what each question asked. Here are 4 of the 7 questions I worked through:

Question 3

This was probably the question I had the most confidence in answering yet I still didn’t really know what was going on. I assumed I needed to use the ratio test and then was able to recall that for some reason you place the resulting x expression into the middle of  –1 < x < 1 and then isolate x. I also remembered that I needed to input the end-points into the OG formula to see if the resulting expression converged or diverged which would determine if the end point was included in the radius-of-convergence or not. (I don’t know if any of what I just said makes sense.)

Question 4

This was the question I cheated on. If you go back to my Week 173 post (which is what I did to cheat) I answered literally this exact question. Before I found the question in my older post, first I had to look up HOW to write out the Maclaurin series for ln(1.4) which in itself was incredibly disappointing since after 3 months, I still couldn’t remember how to do it. Then, after I reviewed how to write a Mac-series, I spent about 40 minutes just trying properly write out the sequence of derivatives for ln(x) which I kept making mistakes on. Even after I finally wrote out the proper Maclaurin series, I still had no clue what was going on so eventually I gave up and reviewed my notes. To be honest, I still don’t know what’s going on with this question. I don’t understand why the Lagrange Error Bound theorem works which is the whole point of this question. 😒

Question 5

Once again I had no idea what I was doing in this question and just happened to get it correct. I assumed I needed to throw a_n into an integral with the lower bound being 1 and the upper bound being b since that’s the variable that was in all the answers, plus I vaguely remembered that’s what I did working through these types of questions a few weeks ago. I simply solved the integral and was left with the correct solution. I’m happy that I was able to solve the integral and use the log algebra to solve ln(1 + b²) – ln(2) but I have no idea how the integral test determines if the given series converges or not, or why.

Question 6

This was the question I got wrong which is ironic because, of all the question worked through, I think I had the best understanding of how to solve this one. In my notes above, I rewrote out the algebra properly but I initially didn’t divide the numerator by 2 which kept it as 1 and so I thought the first term of the series was 1. I think part of the reason why I made this mistake was because 3 of the 4 answers had 1 as their first term so I assumed that was correct.

I realized as I was going through the unit test that I would definitely need to go back through everything from this unit. Despite the fact that I was getting my ass kicked on the test, I was optimistic that I’d have a better understanding of it all the second time through. I decided to watch the video below which I had already seen a few weeks/months ago but thought I might understand it a bit more this time around:

It was the first 10 minutes of this video that helped me understand why T-series work the way they do:

1. The host of the video uses cox(x) as an example. He says that a lot of times using a polynomial that represents cos(x) can be easier to work through in physics questions than cos(x) itself. He then works through how to make a second-degree polynomial that represents cos(x) at x = 0, i.e. a function that looks like p(x) = C₁ + C₂x¹ + C₃x² where C_n is the constant/coefficient in front of each variable. 
2. The first thing you should do it start with p(x) = C₁ = 1, i.e. a horizontal line at y = 1. That way, at the very least, both p(x) and cos(x) both equal 1 at x = 0.  
3. The next step is to add a term to p(x) that represents the slope of cos(x) at x = 0. To do that, you need to find the derivative of cos(x) which is –sin(x) and input x = 0 into the derivative. Since –sin(0) = 0 then the polynomial turns into p(x) = 1 + 0x¹. This means that the slope of cos(0) = 0 which is true since the tangent line at cos(0) is horizontal.
4. Then, to find the rate-of-change of the slope for cos(x), you need to solve the second derivative of cos(x) at x = 0 is –1:
   
   1. d²/dx²[cos(x)] = –cos(x)
      1. –cos(0) = –1
         
5. You then find the second-derivative of the polynomial which corresponds to the rate-of-change of the slope of the polynomial at x = 0:
   
   • d²/dx²[p(x)] = d²/dx²[C₁ + C₂x¹ + C₃x²]
     • = 0 + 0 + 2C₃
       
6. Since the actual rate-of-change of the slope of cos(0) = –1, then 2C₃ = –1 >> C₃ = –½:
   
   • p(x) = C₁ + C₂x¹ + C₃x² 
     • = 1x⁰ + 0 + (–½)x²

Writing out everything above has made it seem a bit more confusing to me, but I think the general idea is that you can take a function and represent it by finding it’s y-intercept (if you use x = 0), then its slope, then the acceleration of its slope, then the acceleration of the acceleration of its slope, etc.

………….

🤔

………

I don’t know what I’m talking about but I think I’m on the right track!!!

I’m hoping by the end of next week I’ll make it through the unit test for Series (1,600/2,000 M.P.) and hopefully (pray to God! 🙏🏼 🙏🏼 🙏🏼) I’ll have a better understanding of why T-series/T-polynomials work. I’m hesitant to say this, but I do think I’m on the brink of finally understanding it all which would be a huge relief. This is the problem with math – I never know how close I am to understanding something until I FULLY understand it. I could be a few days away from figuring it out or it could be a few months. I know I’ll get there at some point but the uncertainty of it is what makes it so hard. 😩