I didn’t get much done this week. 😔 Looking back through my notebook, I wrote 22 pages of notes which is ok but, compared to usual, that isn’t nearly as much as I usually do. I watched most of the remaining videos from Series that I needed to get through and finished the single exercise I needed to do but I didn’t have another attempt at the unit test. I’m pretty disappointed with how this week went but I’m sure I reach my minimum goal of spending >5 hours watching videos and working through questions so it’s not like it was a complete failure. I’m mostly disappointed because this unit is taking me WAY longer to get through than I wanted it to. The silver lining is that this week I got a better grasp on how series work which is helping me understand why they work, so I made some progress even if it didn’t feel like it.
The single exercise I had to get through this week was from the section Ratio Test. I’ve worked on these questions that came up in this section a million times so I didn’t bother screenshotting any of the questions to go through here. The gist of the ratio test is that if a term, (n + 1), is less-than its previous term, n, but greater-than 0 as n goes to infinity, then the series converges. After having done these questions so many times, the algebra seems pretty straightforward. The formula is limn->∞|an+1/an| and if the solution is <1 then the series converges.
One of the things I worked through this week that better helped me understand how series work came from the following video which is a proof as to why a p-series converges when p is greater-than 0 and less-than 1:
To be honest, I still don’t really understand how/why you can shift the bars that represent n to indicate if the function is supposed to be an upper- or lower-bound for the series’ formula. BUT, I do understand how to work through the integral calculus and understand that when p = 1 and when p > 1 the antiderivatives go to infinity as n goes to infinity (i.e. the series diverges). On the flip side, I also understand how when p < 1 the antiderivative goes to 0 since the solution ends up being 1/∞ (i.e. the series converges). Lastly, if p < 0, i.e. it was a negative number, that would result in x going back up to the numerator at which point it would no longer be a p-series. (I just thought about that now but I’m pretty sure that’s correct.)
The last section I worked through this week brought me back to Lagrange Error Bound. When I first got through this section a few weeks ago I made it through by memorizing the formula but didn’t really know what was going on. I’m certain that’s why I got one of these questions wrong on the unit test since I would have forgotten the formula at that point. Coming back to L.E.B. this week, it didn’t seem as quite as difficult to wrap my head around but, even still, I was still a bit confused. I Googled it and found these three videos which helped me get a better idea of how it works:
https://calcworkshop.com/sequences-series/lagrange-error-bound/
(The video in the last link is at the bottom of the page.)
After watching all of these videos the main thing that helped me understand how to solve these types of questions is that M = fn+1(z) and that z is whichever number that’s either equal to or between x and c that makes M as large as possible. (Also, c is the ‘centre’ of the polynomial/function which seems to often be denoted with a.) I’m also now able to visualize that the rest of the L.E.B. formula, (x – c)n+1/(n + 1)!, is simply denoting the next term after the final term of the polynomial. I can also visualize that the polynomial and function look similar when graphed and that the difference/distance between the two at x will be less-than whatever value comes from using the L.E.B. formula. Understanding each of these things has helped me simplify what’s going on in my head but I know that I haven’t fully 100% wrapped my head around it quite yet. That said, I’m optimistic that I have a strong enough grasp on it to get through the unit test this time around, not just because I memorized the formula but because I actually have a decent grasp on what’s going on.
I think I have a pretty good shot at finally getting through the Series (1,820/2,000 M.P.) unit test this coming week. Unlike when I first attempted it (or on my second attempt for that matter) I actually feel relatively confident about my understanding of series now. I know that I still have some gaps in my knowledge of how they work, but for the most part I think I know the majority of the notation and how/why the formulas for series work the way that they do. And as I mentioned just above, I can now actually visualize what’s going on with the terms of different series and can see why they would either converge of diverge as n goes to infinity. Hopefully by this time next week I’ll be done with this unit and can finally move on to the Calc.2 Course Challenge! Not that it will be any easier…