I didn’t pass the Series unit test this week. 😡 I had one attempt at it and got through it relatively quickly but got four questions wrong. I felt like I had a better grasp on the questions this week than I did last week which was a relief, but I’m still pretty disappointed that after 14 weeks I still clearly have some pretty big gaps in my understanding of series. 😒 As demoralized as I am, I know that I’m slowly chipping away at it and feel like I’m starting to understand the bigger picture of series and how they work. It’s always hard to know though as until I understand ~100% of a subject or close to it, I don’t understand it at all and have no clue how close I am to actually getting there. My gut tells me I’m close to figuring it all out but it’s hard to say.

I had a few videos and exercises I needed to review at the start of the week which I got through by Wednesday morning. I got through 4 out of 20 (😶‍🌫️) questions on the unit test on Wednesday and then finished it off on Thursday. Here are four of the questions I worked through between both days:

Question 1

One positive thing I found working through this question was that I managed to pretty easily work through the factorial algebra. I used to have a hard time understanding things like (n + 1)! = (n + 1)(n)! = (n + 1)(n)(n – 1)!, etc. It’s not a part of this question, but similarly I used to struggle to think through certain exponential algebra such as aⁿ⁺¹/aⁿ = a which often occurs in these types of questions, as well. Now I find it pretty straightforward to understand too. I do have a hard time understanding why the ratio test in inconclusive when L = 1. I think it’s because if the bajillionth term over the (bajillion + 1)^th term is equal to 1, then the series of terms that far down the x-axis look like a horizontal line at that point (probably not the right way to phrase it but it’s the way I think about it), and if each term is approaching y = 0, then the series would converge but if the terms are creating a horizontal line at any other y value other than 0 than the series would diverge.

(I have no clue if that’s actually true though.)

Question 2

There was nothing particularly difficult about this question but I wanted to add it here to mention that I can now visualize the math in this question and the notation. It turns out that this was the exact question Sal worked through in one of his first videos from this unit. To solve it he drew out two number lines which are what I now visualize. Below is a screen shot from that video where you can see the two number lines for S₆ and S₇. Being able to visualize the number lines for both S₆ and S₇ helps me to understand that they’re the sum of a₁ to a₆ and a₁ to a₇ respectively. 

Question 3

I got this question wrong because I thought that the first term in the series was 1. I’m pretty sure I thought it was 1 because the numerator of the series’ formula is 1. I made a note that what I need to remember to do on these types of questions is write out at least the first term of the series if not the first few terms to be more certain that I’ve got the solution correct.

Question 4

This question was a bit of a confidence booster for me. I couldn’t figure out whether the numerator, eⁿ, or the denominator, n², would grow faster as n approached infinity. I spent about 10 – 15 minutes looking at my computer screen trying to think it through. I tried to do some algebra using ln(eⁿ) but couldn’t figure it out. I felt like an idiot, like I was making no progress understanding math after 3.5 years of working on it. (By the way, this week is the EXACT 3.5 year mark! 😳) Right when I was about to give up and just guess I asked myself which factor would be larger if n was 100. Thinking of e multiplied by itself 100 times, it was pretty clear to me that it would be larger than 100². Coming to this realization in itself wasn’t the big confidence booster. It was that this turned out to be LITERALLY the exact same way that the KA answer solved the question! I was pumped that I was able to figure it out the same way they did and made me think that I’m actually making some progress and thinking like a mathematician. 🤓💪🏼

After I got through the unit test, I spent the rest of the week reviewing videos and working through two exercises. It was all pretty straightforward and so I only made a note on of one of the questions I worked though:

Question 5

This type of questions relates to the second question from this post. In order to find a_n, you use the same approach as in the second question, but since you’re not asked to find a specific term you subtract S_n–1 from S_n which is the same thing as when I subtracted S₇ – S₆ in the second question. I find this question a bit more difficult to think through, however, since you’re using the variable n in place of an actual number. This makes the algebra seem a bit trickier to me but, nonetheless, it’s not too difficult and much easier to think through now that I can visualize series as number lines.

I finished this week by going through and redoing the exercises I had to do in order to bring the scores for each exercise back up to 80%. I’ll therefore be able to restart the unit test right at the beginning of this week and will hopefully (pray to the lord 🙏🏼) be able to finish off Series (1,900/2,000 M.P.). As I mentioned, I’ve been working on this unit for 14 weeks, i.e. 3.5 months, and have been learning math for the past 3.5 years! I think it’s about time I finally get through this unit and with it Calculus 2. I’ve started thinking that my unofficial goal is to finish the rest of the calculus units by the four year mark of working on KA. Even though that’s six months away, considering how long this unit has taken me, if I want to reach that goal I think it’s definitely time to get through this unit!