This week I got sick on Tuesday and have been feeling like crap ever since. I think I finally turned a corner this morning which is good, but in total I only got ~1.5 hours of work done on KA this week which added up to 1 exercise and 2 videos. 😔 Even though there’s not much to report on, I didn’t want to not make a post and ruin my 169-week streak of making a blog post every Monday but, considering I got such little work done and that my head is still spinning, this isn’t going to be much of a blog post.

The one exercise I worked through this week was from the section Absolute and Conditional Convergence. My understanding of what was taught in this section is that there are alternating series that converge on their own but if you place absolute value brackets around the formula, they diverge. Here are 3 questions I worked on from the exercise:

Question 1

This is a good example of how I still haven’t fully wrapped my head around all of these convergence/divergence tests. In this question I figured out that the numerator ln(n) would be larger than the denominator n as n went to infinity, and that the series would go to 0 regardless of whether or not there were absolute value brackets around the formula so I assumed it would converge absolutely. As you can see from KA’s answer, however, since ln(n) > 1/n and 1/n diverges, ln(n)/n also diverges which is something I forgot to check.

Question 2

Question 3

The next section I worked through was titled Alternating Series Error Bound and had to do with finding the value of the “remainder” after a certain term within a series. Here’s a screen shot from the first video in the section:

In this example, Sal was using the formula (-1)ⁿ⁺¹/n² which you can see he then calculated the first 8 terms for. What the video goes on to explain is that, in an alternating series, if you find the value of the first x number of terms, then the value of the remaining terms from (x + 1) to infinity will be less-than the value of (x + 1). There’s obviously more to the explanation than that but right now my head is spinning all over the place and I don’t have the energy to try to explain why that’s the case.

This coming week I’m hoping to make a lot of progress on KA to make-up for this past week. I’m 48% of the way through Series (960/2,000 M.P.) which seems pretty good except I still have 33 videos left to get through… There are only 7 sections left in this unit but a bunch of them have 6 or 7 videos in each section. The next section I’ll be working on is titled Taylor and Maclaurin Polynomials Intro which I’m interested to start on because I’ve heard the name Taylor Series a few times before but I have no clue what they are. Hopefully I’ll be feeling back to normal in a day or two so I can get back on track. 🤧