Reminders before Final Exam

Please be reminded that the lecture notes from Week 8 up to the end are contained and updated in these two files:

Notes for Sequence and Series (Week 8-10)

Power Series, Taylor and Maclaurin Series (Week 11-12)

Also, you should check out the files that I uploaded to the shared folder I emailed you all. Just reading it a couple of days before the exam has very limited use. Try to work out the problems yourself and use my answers just as reference and hints.

Good luck in all your exams! Farewell!

Posted in Lecture

Lecture #23 – second to final lecture on course materials

So we did cover most of the final part of the notes (see previous post for the file) except the last two examples showing how to prove a Taylor series converges to the function it represents (as we always expect and assume), within the interval of convergence.

The more important parts of the examples are how to estimate the error of Taylor polynomials  (partial sums of the Taylor series) using the Taylor inequality for the remainder term and the Alternating Series Estimation Theorem (ASET) if the power series is alternating.

We will finish off these examples next Tuesday and start reviewing for the final exam in both lectures next week. Those who need an extended period of social isolation to review stuff could safely skip the lectures and travel to the Arctic.

The pace and difficulty will be set for the purpose of just passing the course. It should feel leisurely for people who have been following well.

Posted in Lecture

Final set of notes uploaded

The file taylor is now updated to cover everything I want to teach for Taylor and Maclaurin series.

The marathon is coming to an end soon.

I will further double check the typed notes we have so far and upload a revision before the term really ends (e.g. make sure my notations agree with those found in the textbook etc…)

Posted in Lecture

Quiz 4 Quick Comments

1. If you still find yourself still arguing about how the general term \( a_n \) is going to zero, instead of the nature of the general term, like:

  • the general term is exactly the form of some known series (geometric and p-series)
  • the general term is a difference (telescoping series)
  • the general term is a positive decreasing function of n, that is integrable (integral test)
  • the general term is alternating with absolute value decreasing to zero (alternating series test)
  • the absolute value ratios of consecutive general terms is converging (ratio test, which apples to alternating series too)

Then be warned that you’re doing something fundamentally wrong. Let me repeat again,

$$ \lim_{n\to\infty} a_n = 0 $$

says nothing conclusive whatsoever for your series under investigation (the only “exception”: unless you couple it with the knowledge that the series is alternating and absolute values of terms decreasing). Arguing the general term to somehow converge to 0 and quoting “whatever test”, all these amounts to a zero mark in a harsh marking scheme. You know I am harsh against irrelevant details.

It is only when the general term DOESN’T converge to zero that the series must diverge by the Divergence Test (Or Term Test if you read sources other than the textbook). In EVERY case when the series does converge, all efforts into arguing that the general term sequence is going to zero (in whatever vaguely described ways) are going to be in vain. You need to make explicit comparisons or explain the nature of the general term as above to get credit.

2. The comparison tests require your knowledge of another series \( \sum_{n=1}^{\infty} b_n \) which you know to converge / diverge. If you guess divergence, get a series below: \( b_n \leq a_n \), if you guess convergence, get a series above. The required conditions for the direct comparison test is that the sequences are positive and the series you use for comparison is known to diverge / converge. The limit comparison test is easier to use, you get another series so that the ratio of terms converges to some nonzero number:

$$ \lim \frac{a_n}{b_n} = L \neq 0 $$

and of course not infinity (i.e. it diverges). Then either both converge or both diverge.

Posted in Quiz & Midterm

Quiz 4 Solution.

Lecture Notes are already posted, no major changes.

Hi everybody

For solutions to today's quiz, see

HW9, Section 11.3
Q1: Q9
Q2: Q25
HW9, Section 11.4
Q3: Q31

HW10, Section 11.6
Q4: Q17
Q5: Q27

I am not posting separate solution sets. Check previous email below
for the location of the solutions.

Posted in Lecture, Quiz & Midterm

Lecture #22

Notes already uploaded, by expanding the previous file taylor .

Most probably we won’t be able to cover everything shown. It can be helpful to your note taking if you print this file out and bring it along.

Posted in Lecture

Lecture #21

Today we finished the last example on Ratio Test and introduced Power Series:


As suggested by the name of the file, our ultimate goal is to understand Taylor Series representations of functions.

Posted in Lecture

Lecture #20

Today we covered Alternating Series again and then most of the examples on Ratio Test. (Section 11.5-11.6)


The only example that is missing bridges Ratio Test to the next topic: Power Series (Section 11.8)

The combined file:


has everything in the Sequence and Series section, but it is constantly updated to correct typos, add examples or exercises. Expect it to be finalized before the end of term for your review for finals.

Posted in Lecture

Notes for Sequence and Series (up to Section 11.6, Ratio Test)


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Lecture #19

For some reasons, I didn’t have the book with me and I wrote the notes on top of my head, (plus the sleep-deprived condition after MT II marking…), then as expected, I made a couple of mistakes, including a really bad one of omitting to check the absolute values of the terms in an alternating series SHOULD BE MONOTONICALLY DECREASING before concluding with the Alternating Series Test.

I’ll cover the Alternating Series Test again on the coming Tuesday. The corrected notes is here:


Posted in Lecture

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