I finished the unit test for Series this week getting 15/20 correct. Even though that’s probably one of my worst scores at a unit test, I was actually somewhat happy with how the unit test went. My intuitive understanding of what was going on in most of the questions was better than what I thought it would be, even though I got a bunch of questions wrong. On the questions I got wrong, I had a general idea of what was happening but I didn’t know what the formula was or, if I did, how to properly use it. This indicates to me that I don’t have a full understanding of why these formulas work but I can tell I’m getting closer. By the end of the week I was able to get through the 5 exercises I needed to finish in order to bring my score in each of those sections back up to 80%. I’ve mentioned in a few other posts that you need to have the score for each section be 80% before doing this unit test otherwise, even if you get a perfect score on the unit test, the score for the entire unit won’t be 100%. All that to say, I’ll be kicking off next week with the start of the unit test and am feeling pretty good that I’ll have a chance at making it through with a perfect score on my first attempt. ☺️
(Longest opening paragraph ever?)
The following are six questions I worked through this week. The first four are from the unit test and the last two are respectively from the section Ratio Test and Representing Functions as Power Series.
Question 1
In this question I couldn’t recall what I needed to do to find the solution. Even after looking at the answer given by KA, I vaguely remembered the formula used for this question but don’t really understand why it works. The silver lining is that I was able to work through the algebra pretty easily once I was shown the formula. Although that doesn’t seem like much of a silver lining thought if I’m being honest. 😔
Question 2
I got this question wrong on the unit test but actually did the algebra correct, just didn’t understand the first part of the question where it asks about onvergence/divergence. When I went back to redo the exercise that this question came from, I watched a video from that section and think I figured out why a series converges if it’s < 1, diverges if it’s > 1 and why the ratio test is inconclusive if it equals 1. I believe it’s for the same reason that when r < 1 a series converges and when r > 1 it diverges. In fact, I think it’s literally the same thing in that if the (n + 1)th is less-than the nth term, than the ratio between the two (i.e. r I’m pretty sure) is < 1 which indicates the series will converge. Ditto goes for when (n + 1)/n > 1 in that the series will then diverge. (I think…)
Question 3
The page of notes I took for this question were notes that I redid after I already got this question wrong. The reason I got it wrong was partly because I used the wrong series to compare the series in question to. I thought I needed to compare the series in question to 1/n2 but now I don’t really know why I thought that… It seems pretty obvious to me now that the square-root cancels the power of two in the series in question, 1/(n2 + 1)1/2, and that the +1 becomes irrelevant as n approaches infinity. One thing I would have gotten wrong regardless if I had chosen the right series to compare it to was the algebra needed to solve this question. I don’t think the algebra is that hard to work through but I forgot that when using the comparison test, you need to divide the comparison series by the series in question and find the limit of that expression as n goes to infinity.
(Pretty sure non of that made sense.)
Question 4
This was another question that I had an idea of how to find the solution but couldn’t remember the last few details. I correctly remembered that I needed to put the series formula into an integral and solve it which left me with 1/(3n * 9), but I didn’t know what to do after that. What you need to do from there is recognize that you can separate that solution into two expressions, 1/9 * 1/3n, and then you can also turn 1/3n into (1/3)n. Then you have to know that these expressions, 1/9 and (1/3)n, can be thought of as a and r which are the first term of a series and the common ratio of the series, respectively. Since r = 1/3 < 1, you then simply input those values into the geometric series’ formula to determine what the series converges to. That said, I still don’t understand how/why the integral of f(x) from 0 to 1 is apart of this question…
Question 5
This question was the exact same question as question 2 from and uses the same formula/algebra to solve.
Question 6
This question was pretty easy to work through was good practice using factorials cancelling factorial factors. To solve this question, you have to first understand that you need to take the series formula and find its second derivative with respect to x. To do so, you also need to understand that when finding the second derivative of a series’ formula with respect to x, you take out the expressions that contain n and don’t contain x. After doing this I was then left with (–1)n(2n + 1)(2n)(x2n–1)/(2n)!. Since (2n)! implies (2n)(2n – 1)(2n – 2)… then the (2n) in the numerator and the the (2n) in the denominator cancel with each other and you’re left with (2n –1)! In the denominator. Working through these types of questions has made factorials much less intimidating and easier for me to understand and think through.
Like I said at the beginning, I’ll be starting the unit test for Series (1,900/2,000 M.P.) right at the start of this coming week. If I don’t end up passing it on my first attempt, I’m optimistic that I at least won’t get too many questions wrong so I’ll (hopefully) be able to go back and get through the required exercises quickly and have a second attempt at it before the end of the week. Come to think of it, I really have no excuse not to pass the test before the end of the week. Boom. That’s my goal. If I manage it I’ll finally be on to the Course Challenge for Calc.2 which, if I’m being honest, I’m a bit intimidated/scared to start. I’m going to have to get through it eventually so better soon-than-later!