This week I got through ~3 sections in Series but that only ended up being 4 videos and 3 exercises. Just about everything I worked through this week had to do with testing series to determine if they converge or diverge. The first 3 sections I made it through were titled Comparison Tests, Alternating Series Tests, and Ratio Test. I also made it through the single video from the final section I worked on which is titled Absolute and Conditional Convergence (which thankfully didn’t have to do with any type of convergence or divergence test). I think I’m definitely getting a better handle on the notation used for series, in general, and feel much more comfortable with them now. It seems like I’ve learned about a million different ways to test series over the past few weeks so I’m having a hard time remembering the nuance between all of them. In any case, although I only made it through a handful of vids and exercises this week, I do feel like I’m making decent progress through this unit which I’m happy about. 😬

I said at the end of my post last week that this week I’d give a better explanation for the difference between the Comparison Test and the Limit Comparison Test. I only worked on one exercise from that unit this week, however, and I didn’t get much of a better understanding of the difference between the two tests so I unfortunately can’t give a better explanation of their differences this week.

The following section, Alternating Series Tests, was much easier to understand. The gist of an alternating series is that the (+) and (–) signs alternate between each term in the series. These types of series converge if they do two things:

1. As lim_n->∞, the formula a_n goes to 0, and
2. The sequence of a_n is always decreasing.

Respectively, here’s a screen shot from the second video in this section a page from my notes where I worked through it, and then an example question from the exercise with a page form my notes that explains the main part of the notation used for alternating series:

Question 1

The next section I worked through, Ratio Test, I’m still finding a bit difficult to understand. I think the idea behind this type of test is that if you have a formula for a series, a_n, and as lim_n->∞ of a_n+1­/a_n = 0, then the series converges. Here’s a screen shot from the first video in this section and my note from that video below it:

As a side note, this was the first time that I can remember working through this type of algebra that involved factorials. I didn’t find it too difficult to understand but, since I’d never done it before, it took me a minute to wrap my head around. 

Here are two questions I worked through from the exercise:

Question 2

Question 3

As I mentioned at the beginning, I finished the week working through a section titled Absolute and Conditional Convergence which I spent all of Saturday not being able to understand. Thankfully, I think I finally figured it out on Sunday morning. The gist of what it talks about (I’m pretty sure) is that there are certain alternating series that will converge on their own but if they’re placed inside absolute value brackets, they will diverge. There’s only one video and one exercise in this section so I’m going to save it until next week once I’ve gotten through the exercise and am (hopefully) more confident in my understanding of it.

I’m now 44% of the way through Series (880/2,000 M.P.) but I’m less confident that I’ll be able to make it through the unit before the end of the year. There are still THIRTY-SIX videos left for me to watch (😳) and 9 exercises left to get through before I can start the unit test. I have a feeling that the unit will get harder as I get closer to the end of it, as well. Nonetheless, I’m still going to grind away and hope that I can somehow get through it all before the start of 2023. Not the end of the world if it doesn’t happen though.