Before sitting down to write this post, I felt like I had a pretty pitiful week on KA. Looking back through my notes however, I managed to get through 3 sections in Series this week, apart from finishing one last exercise in the third section. The things I’ve been working on in the past few weeks have seemed very obscure and I’m finding it hard to sit down for a prolonged period of time and study. In the past when I’ve been working on things that I’m able to understand relatively easily, I found it much easier to concentrate for an hour or more whereas the past few weeks have felt like an absolute grind to concentrate. I’m hoping that I’m over, or getting close to, the ‘hump’ of learning the definitions, notation, main concepts, etc. of what I’ve been working on in Series so that soon it all seems a bit more intuitive. 🙏🏼 😰
I started the week off on the first video from a section titled Integral Test. As you’d expect from the title, this section explained how to use integrals to determine if a function converges on a number or diverges to infinity. Here are a few pages from my notes that go through how the integral test works:
The key things to remember in order to use the integral test is that the formula for the terms (which seems to be denoted with an following Σ) has to be 1) continuous, 2) positive and 3) decreasing from [k, ∞) where k is the first number in the index. (I have no clue if I’m saying this properly…) You then state that an = f(x) and find the integral of f(x). Here are two examples from the exercise I worked through in this section:
Question 1
To be honest, I don’t really understand the bottom part of the KA explanation. I know that as x gets bigger the numerator gets more and more negative but also that the absolute value of the denominator grows much faster than the numerator so the expression overall gets smaller and smaller as x goes to infinity. There’s no point from 2 onwards to infinity where the expression becomes positive, however, so this means that f’(x) is always negative, i.e. f(x) (a.k.a. an) is always decreasing from 2 to infinity. Therefore, the integral test can be used.
Question 2
On Wednesday I started the following section titled Harmonic Series and P-Series which only took me a day to get through. The Harmonic Series is a famous series that equals n=1Σ∞ 1/n, i.e. [1/1 + 1/2 + 1/3 +…]. It got its name from music as when the sine waves of musical notes that fit neatly into one and other (i.e. you can get two ½ waves into 1/1 and three 1/3 waves into 1/1 etc.) they create a harmony. (That’s a terrible explanation but if you Google it you’ll easily be able to understand what I’m talking about.) A P-series is any series where the formula is 1/np where P simply stands for Power. Here’s a page from my notes that talks about both series:
The key thing I learned in this section is that if P is greater-than 1 the series converges (because each value in it gets smaller and smaller since the denominator gets larger and larger) and vice versa if P is greater-than 0 but equal-to-or-less-than 1.
The last section I worked through this week was titled Comparison Tests and it was the section I struggled with the most to understand. Here are three pages from my notes that goes through the general explanation of a comparison test:
There’s a difference between what’s called a Comparison Test and a Limit Comparison Test but but I don’t really understand the latter. Since I don’t really understand what’s going on, I’m not going to bother trying to explain it in this post but I’ll try again in my post next week. Nonetheless, here’s an example of how the Limit Comparison Test works:
Overall, I do feel like I’m starting to get a better handle on series, in general, but at the same time I feel like I’m not even past the halfway point of understanding what’s going on. I’m only 32% of the way through Series (640/2,000 M.P.) so I still have a long way to go in that regard. Like I said earlier, hopefully I’ll reach a point where it all starts to click and things seem less overwhelming and difficult to grasp. One thing I’m reminding myself is that this is the LAST UNIT in Calc.2! So, I just need to keep grinding and eventually I’ll reach the light at the end of the calculus tunnel. 🥵