Date Range: October 27 – October 31
Hours Spent: 15
- 5 hours on lectures Professor Leonard – Calculus 2 Lecture 9.1: Convergence and Divergence of Sequences
- I watch the lecture and scratch out notes, then after I’ve finished I go slowly through my notes and rewrite them neatly into a different notebook. This second notebook is kept for longer-term reference. This also helps the lecture sink in
- 10 hours on exercises (Thomas 13th ed, questions 1 – 74)
Summary:
I’ve begun a new unit on infinite sequences and series. Herein was an introduction to what sequences are, how to write them, the terminology, etc. I first am taught to look at a formula and calculate a series {a1, a2, a3, … an } , then determine a sequence around a finite number of fixed terms. This is wading slowly into the water. We learn about the properties of sequences and the limits of sequences. Limits are key here, as we’re ultimately confirming convergence/divergence. We learn about monotonic sequences and bounding as it relates to convergence/divergence. We apply the Squeeze Theorem to determine convergence/divergence, and are introduced to six commonly occurring sequences.
Personal Observations:
- Many of the problems were self-evident – I understand how limits work and can see quickly whether a sequence converges or not.
- It’s cathartic looking for patterns. It’s a beautiful day, the sun is shining in, I’m snacking on cashews while listening to Back and looking for patterns in sequences – bliss!
- Playing with recursion is fun, if not mind-bending at times. I’ve worked some with recursion in coding, but never spent any considerable time number crunching recursive series manually (mentally)
- As the series involves exponents and natural logs, the problems get trickier. I encounter lots of indeterminate forms, L’Hopital’s, etc. For example:
- Like with other tricky Calculus problems, I find myself revisiting themes I’ve not yet memorized, like rules and patterns of natural logs, mixing them up with other Calc tools – there’s so much to remember. Or, more accurately, there’s so much that needs to be drilled to the point that I can immediately recall and apply it.
Reflections:
When I think about how much I’m learning, I can’t help but reflect on the past, on foregone studies. Although I don’t think I’d ever have appreciated learning in the same way, I believe my life would be drastically different had I followed through on higher learning in my earlier, formative years. I think about how many interesting pursuits a strong math foundation will enable – and could have enabled. Well, the best time to plant a tree was 20 years ago. The second best is today.