I got rocked this week and didn’t get close to hitting my goal of getting through the Series unit test. I almost felt like I went backwards. 🤬 I only made it through the unit test once and got 9 questions wrong. I’m pretty sure the questions I got on my first attempt last week were easier than the questions I worked through this week so I was caught off guard and realized I still have a lot of things to learn. That said, it wasn’t all bad. I realized a few key insights into series and how they work throughout the week. Up to now, I haven’t been able to visualize what’s going on with series in my head, but this week I think I made some progress in that regard. (I think.) I’m pretty frustrated and demoralized at this point but am certain that I will figure this all out eventually, it’s just taking me way longer than I thought it would.

Here are four questions from the unit test that I thought were worth talking about:

Question 1

As you can see, I got this question wrong, but this question gave me two of the biggest insights I’ve had into series and how they work. This was the first time I plotted each term’s value on a graph. Since S₁ has (–1)ⁿ in it, as n goes to infinity, each subsequent term alternates above and below the x-axis. Each term gets closer and closer to either 1 or –1 but will never actually reach either. In S₂ the series doesn’t alternate and each term gets closer to 1, again, without ever actually touching it. I got this question wrong partly because I thought that since both series converge to 1/–1 then that means both series converge, in general. This isn’t true, however, since the concept of a series is that you continuously add each subsequent term, and therefore if you add 1 forever S₂ will definitely diverge. I’m a bit confused about S₁ though since as n goes to infinity it would seem to me that the positive values of n that approach 1 would cancel with the negative values of n that approach –1. 🤔 In any case, plotting the values of each term from both series REALLY helped me to be able to visualize what’s going on with terms from different series. I can also visualize why n must approach 0 for the series to converge. (I think of it like a function that goes to 0 as x goes to infinity where you could then find the area from a lower bound to infinity [i.e. as x approaches 0] at which point the area would converge to a certain number.)

Question 2

This was a pretty straightforward question that I got wrong because I didn’t put it together/remember that taking the square root of 4 can equal –2. I thought that I needed b’(x) to equal 2 and then I had to pull a –1 out of the integral. I thought the answer was actually –arctan(2x) which wasn’t an answer given so I assumed it was –arctan(–2x) and obviously got it wrong. One positive thing from this question was that I was quickly able to recognize that the expression in the integral was the derivative of arctan(x) and knew the algebra I needed to solve it apart from forgetting that 4x² could be (–2x)².

Question 3

This was a question I didn’t understand and ended up getting it wrong. Even after looking at KA’s answer though, I’m still pretty confused. Since a_n = (–1)ⁿ (2/n)^p, the series alternates. Taking the (–1)ⁿ out of the formula leaves you with (2/n)^p which I think is the main part of the expression you need to think about to solve this question. Since (2/n)^p = 2^p/n^p than it seems to me that if p = 3/2 then the numerator, 2^p, would equal ~2.824. This seems like the series would then converge on positive and negative ~2.824 which, from what I worked through on question 1 above, means the series doesn’t converge. If I’m right about that, KA’s answer states that p > 0 but I think the answer maybe should be 0 < p < 1. I’m not confident in that but that’s what it seems like based on what I worked thought in the first question.

Question 4

This was another question that I just straight up didn’t understand, the reason being I didn’t know what the notation was talking about. I didn’t realize that a_n means something like, “the value of a given term, a, at some point in the index, n.” Even if I had known what a­_n meant, I don’t think I would have been able to think through and figure out the formula. 

I finished the unit test on Thursday and found out I only had to redo one exercise before I could redo the unit test. This was a bit surprising to me considering how many questions I got wrong. There were a bunch of videos that were starred (i.e. recommended) for me to watch for me to rewatch. I decided that although I only need to redo one exercise, I’d go through all the recommended videos and make notes on them so that hopefully I can actually understand what’s going on. I made it through the 6 videos and took a few notes from them. Here’s one of my notes that highlights some of the notation used in series:

This note just above was related to question 4 from further above and helped to understand the difference between S_n and a_n. 

Here’s a note I took from one of the videos that was titled n^th Term Test:

This video had the same type of question as question 1 from up top. My explanation at the bottom of this note was one of the key insights I had this week which I mentioned earlier. 

Lastly, here are two screen shots I took from two videos in the section titled Integral Test:

I tried to understand what was happening in these two videos for more than an hour and still don’t really know what’s going on. I don’t understand why in the first screen shot when Sal uses the integral test to determine if the formula 1/n² converges he sets the function 1/x² as an upper bound and draws each term as a left-Riemann sum, whereas he draws n as a right-Riemann sum in the second video where he works through the formula 1/n. I looked at the comments of one of the videos and saw that someone had the same question:

As you can see from my comment, I don’t really understand how you can just decide whether to use the function as an upper- or low-bound of the series. But I’m pretty sure that’s what you do…

I still have nine recommended videos left to watch before I’m going to redo the unit test. Hopefully I’ll be able to get through them early this coming week so that I can finally get through Series (1,790/2,000 M.P.).  I’ve said this many times before, but as much as I’d like to get through this unit this coming week, I’m really hoping and would much rather get a strong grasp on series and how they work. I’m hoping will happen as I go back through the remaining recommended videos. I think it might to a certain degree since I’ll be rewatching them with a slightly better understanding of what’s going on this time around which I think will make it more likely that I’ll connect certain dots that I missed initially.🤞🏼