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!

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.

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…)

• 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.

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.

taylor

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

wk10day20

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

The combined file:

notes_seq_series

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.

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

wk10day19