I should go back and see how many weeks in a row I’ve said something along the lines of, “my progress this week was very slow.” This week was more of the same. 😔 I made it through 4 videos and 1 exercise but am still struggling to understand what the hell is going on. I can’t visualize what a series looks like on a graph when it converges which is the big part of the problem. The silver lining is that every so often I’m given some tricky algebra that would have crushed me a few months ago and I’m able to work my way through it. When this happens it reminds me that even though I’m having a hard time understanding series, I’m definitely still making progress in learning math, in general. In any case, since I only made it through the 1 exercise this week and didn’t really have anything new sink in, this post is just going to be 3 example questions that I worked through. Better than nothing I guess. 🤷🏻‍♂️

The exercise I worked through was first of two exercises from the section Representing Functions as Power Series. The questions from this exercise (i.e. the ones below) are a good way to highlight what I said up top, that I don’t understand how a series can be equal to a function. A function is continuous but a series is a list of terms. When you plot a series on a graph, say on Desmos, it seems to me that each term, n, would be a point on the graph at each positive integer whereas a function would be a continuous line across the graph. AGH!!! I don’t understand… Anyways… Here are three of the questions I worked through:

Question 1

Although I don’t understand the ‘why’ of what’s going on in these questions (i.e.  I don’t understand why you would do this math, why it works, or what the hell’s going on in general… 🤬), I was able to figure out the ‘how’ (i.e. how to work through the algebra, woo! 🥳). To find the second derivative of f(x) from the given series, you take the formula, a_n = (–1)ⁿ * x²ⁿ⁺¹/(2n)!, and find its second derivative. One of the keys to do this is to understand that n can be factored out when working through the derivative calculus in the same way that any coefficient can be factored out. This is because (as far as I understand) you have to find the derivative with respect to x and not with respect to n. (To be honest I don’t really know what that means but I’m pretty sure it’s right.) As well, as you can see from my last note, to solve these types of questions I needed to use a lot of algebra to simplify factorials. This was helpful for me because I was having a hard time visualizing how factorial simplification worked so it was good practice.

Question 2

This second question was similar to the first except that I was asked to find the integral of the series’ formula instead of its derivative from x = [0, 1]. (Don’t know if that’s the right way to say it or if I’m making any sense at all…) I found this pretty straightforward, especially since the lower bound was 0 which meant the second term in F(a) – F(b) was non-existent. This question was useful for me to practice working through integral calculus since it’s been a while since I’ve done that, plus I found the algebra used in the final few steps of this question helpful practice too.

Question 3

I forgot to mention that it took me a few days to get through this exercise and it was mostly because I couldn’t understand the type of question just above. My initial understanding was that in the type of question above I could simply find the 3^rd derivative of a_n and then input x = 0 into the 3^rd derivative, but when I tried doing that it never worked. I may have just been doing the math wrong, but as you can see from the KA answer, that’s not how they solved the question so I have a feeling that solving a_n for its 3^rd derivative wasn’t the right thing to do. Once I started to solve these questions by writing out the first few terms of f(x) and then finding each terms’ 1^st, 2^nd and 3^rd derivatives, the math became easy to work through but was pretty slow to get through. 

There’s only one more exercise left to go in the section Representing Functions as Power Series. I tried working through it at the end of this week but failed pretty miserably. Since I don’t understand the questions from that exercise I’m not going to try and bother explaining anything from it, but here’s a screen shot from the video that related to the questions from that exercise:

Like I said, I don’t really know what’s going on with these questions but it has something to do with writing out the first few terms of a given series and then either finding the terms’ derivatives or antiderivatives to match the formula or the terms from another series. I got rocked trying to figure out these questions this week but hopefully by next week I’ll be able to explain how they work.

I’m guessing the tone of this post has felt pretty negative but the good news from this week is that I only have that final exercise left to go in Series (1,520/2,000 M.P.) and then I’ll finally be able to get started on the unit test. I’m almost certain that I’ll end up having to work through the unit test at least 2 or 3 times, but hopefully in doing so that will help everything I’ve learned seem a bit more clear. Even though it’s taken me a long time get to the end of this unit and I’m having a hard time understanding everything, I’m pretty proud that I’ve finally made it to the end especially considering that this is the LAST UNIT of Calc.2! I mentioned this a few months ago but there are still some more calculus units I’ll need to get through after this course, but I feel like I’m on the final stretch and that I can see the light at the end of the tunnel. If I can get through all the calc units before the start of September I’ll be happy. And it will have only have taken me four short years! /s