During one of my calculus classes in university, we were running behind in terms of class content. Within the last few days of class, the professor announced that the topic of Taylor and McLaurin series and expansions weren’t going to be part of the final exam, but he would hold an optional “extra” class for those who were interested.
I was interested, but I also live far away from the university, which meant I didn’t want to drive a total of eighty minutes for fifty-five minutes of class. So I didn’t go. I figured it wouldn’t be a big deal, since it was just a small topic in my calculus class.
Unfortunately, I was dead-wrong. As any physics student knows, being able to write a function as a power series is a very useful technique to know, and is used all over physics. The reason is simple. It can be difficult to solve the differential equations that are encountered in physics, and using a power series expansion can allow us to solve problems to a great approximation. Knowing how to use this technique to express a function as a polynomial is powerful.
Because I skipped that class, I didn’t get to see this topic in detail. As such, I continued through my education with only a rough idea of how it worked. This meant that when professors would tell us to expand a function as a power series in order to solve a problem, I would always be slightly behind, not quite sure how to do it. I knew that it involved derivatives and factorials, but it was clear even to myself that I wouldn’t be able to do it on my own.
The problem here is one that many students face. They end up not really understanding a concept, or miss it for whatever reason during the semester, and then go on with their education with this missing gap in their knowledge. That’s fine, but these gaps do show up later on. At that point, it’s usually more annoying to go back and learn the concept, so students either try to fake their way through understanding, or fail.
It doesn’t have to be this way. I’ve decided to put in some time to look at expanding a function as a power series, because I know that it’s an important skill that I need in my toolbox. It won’t be something I’ll figure out in five seconds, but it’s a good investment of time.
I am sure these same kinds of weaknesses exist for you. Perhaps there was a concept that kind of “slipped through the cracks” for you when you first came across it, and you just haven’t thought about it in a while. These are your weak points, and it takes honesty to admit that they are there. Furthermore, it takes a certain amount of willingness to say, “I’m not satisfied with acting like I know this. I want to really understand it.” It’s not easy, but it’s important. I will keep on beating the same drum: in mathematics and science, concepts build on top of each other. If you don’t have a strong foundation, it is difficult to learn about new concepts. It’s possible, but your understanding will be riddled with holes. If you don’t believe me, find a topic that you know next to nothing about, and then find some lecture notes or a textbook aimed for an advanced audience. I’m willing to bet that almost none of it will make sense to you. That’s because you don’t have the foundational experience necessary to jump into these resources.
If you want to get better at anything, it’s crucial that you identify your weaknesses, and then work to improve them. This latter part is just as important as the former. Saying you have weaknesses is one thing, but working hard to address them is a different challenge. However, if you are willing to put the work in, it is doable.
Don’t do like I did. Don’t miss a concept and let it go ignored for years. It will come back to haunt you, so you might as well put in the work to understand it.