Unit 13: Series

The two slides work well together or individually (any of them).

• Slide 1 is an example where students can compute a series from beginning to end using the definition.

• Slide 2 reminds students that we cannot assume infinite sums behave like finite sums. Series are a new concept and properties cannot be taken for granted.

Answers:

• Slide 1:   $\displaystyle S_{k}= \frac{1}{2}\left[ 1 + \frac{1}{2}- \frac{1}{k+1}- \frac{1}{k+2}\right]$

• Slide 2:   The series is divergent to $-\infty$. The split we did as difference of two series was illegal, because it was “$\infty - \infty$”.

Simple question to hammer on the definition of series and of partial sums.

Answer: $\displaystyle \sum_{n=1}^{\infty}(2n+1)$

This activity foreshadows (slightly) positive series and comparison tests.

It also insist on the definition of series as a limit of partial sums.

Answers:

• 1 - we do not know: every series satisfies this

• 2 - convergent: the sequence of partial sums is increasing and bounded above

• 3 - we do not know

• 4 - divergent: the series grows faster than $\displaystyle \sum_{n}^{\infty}M$

This question foreshadows comparison tests.

Students will later learn that the harmonic series diverges using the Integral Test.

This question is an attempt at doing an “elementary” proof from the definition of series.

How I use this question:

• I give students a bit of time.

• We discuss together the answer to 1.

• I give students more time for the rest.

Answer:

• $\displaystyle r_{1}= 1$, $\displaystyle S_{1}=1$

• $\displaystyle r_{2}= \frac{1}{2}$, $\displaystyle S_{2}= S_{1}+ \frac{1}{2}$

• $\displaystyle r_{3}=r_{4}= \frac{1}{4}$, $\displaystyle S_{4}= S_{2}+ \frac{1}{2}$

• $\displaystyle r_{5}= r_{6}= r_{7}= r_{8}= \frac{1}{8}$, $\displaystyle S_{8}= S_{4}+ \frac{1}{2}$

• ...

This activity should be easy. It focuses on the definition of a series as the limit of the sequence of partial sums, together with properties of sequences.

Nevertheless, do not rush it! Students can answer this question, but it takes them time to wrap their heads around the notation and get comfortable with it. It is worth it to give them enough time to think through it and discuss it with their peers.

Answers: 1, 3, 4, 5, 6, 7 are true. 2 is false.

Short question to recognize geometric series even when they are “in disguise”.

Works well as a warm up.

Answer: 1, 2, 3, 4 are convergent. The rest are divergent.

Simple question to recognize geometric series and add them up.

Students find this question accessible and even fun. Give them enough time and they will solve it.

There are some subtleties about Question 6:

• They will forget we need to break this into cases, depending on whether $|x|<1$.

• The videos only introduce $\displaystyle \sum_{n=0}^{\infty}x^{n}= \frac{1}{1-x}$. Some of them learned in high school $\displaystyle \sum_{n=0}^{\infty}ax^{n}= \frac{a}{1-x}$. “Are we allowed to use this other formula?” “Do we need to know it?”

• If you try to reduce this problem to using the basic formula only “are we allowed to do those manipulations?”

Answers:

1. $\displaystyle \sum_{n=0}^{\infty} \left( \frac{1}{3} \right)^n \; = \; \frac{1}{1 - 1/3}$
2. $\displaystyle \frac{1}{2} \sum_{n=0}^{\infty} \left( \frac{-1}{2} \right)^n \; = \; \frac{1/2}{1 + 1/2}$
3. $\displaystyle \frac{3}{2} \sum_{n=0}^{\infty} \left( \frac{-3}{2} \right)^n \;$ divergent
4. $\displaystyle \sum_{n=0}^{\infty} \left( \frac{1}{2^{0.5}} \right)^n \; = \; \frac{1}{1 - 1/\sqrt{2}}$
5. $\displaystyle \sum_{n=0}^{\infty} \frac{1}{2} \left[ \left( \frac{-3}{4} \right)^n - 1 \right] \; = \; \frac{1/2}{1 + 3/4} - \frac{1}{2}$
6. $\displaystyle x^k \cdot \sum_{n=0}^{\infty} x^n \; = \; \frac{x^k}{1 - x}$ if $|x|<1$.

There are absurdly heated debates on the internet over whether $0.999 \ldots = 1$. Read the Wikipedia article https://en.wikipedia.org/wiki/0.999... (opens in a new tab)and in particular the sections “Skepticism in education” and “Cultural phenomenon”.

Your class will be split between firm believers that they are different, confused students, believers that they are equal but who cannot prove it rigorously, and those who know what is going on.

Resist the temptation to rush or to lecture to your students and give them the answer. Proper understanding of what is going on is now within their reach if you give them enough time. Use this activity wisely and you will witness many “aha” moments, things finally clicking, and some minds blown.

After students have solve the question, it is appropriate to have a mini-lecture to explain how a decimal representation is just a way to represent a number, that the proper interpretation of such expansions is a series, and how it is reasonable for two decimal representations to represent the same number (just like two fractions can represent the same number).

Both of the slides serve the same purpose. I use one or the other depending on the time I want to spend.

Answers:

The good:

• These activities serve as an advertisement for why we will spend Unit 14 studying power series.

• If successful, students will be impressed and intrigued.

The bad:

• These activities are hard and take a lot of time!

• Resist the temptation to rush into them unless you have enough time and students are very comfortable with the basics. If you are going to end up explaining the answer yourself, rather than having students discover it, they are not worth it.

• If unsure, solidifying the basics of series is probably more important. After all, none of these questions are “necessary” at this point in the unit. Students will eventually learn this in Unit 14 anyway.

Simple question that anticipates positive series.

Some students mistakenly think that $\displaystyle \sum_{n}^{\infty}(2 + \sin n)$ is an example of positive series that is “oscillating”.

Remind them that “oscillating” means “divergent, but neither $\infty$ or $-\infty$”, and not simply that the terms alternate between increasing and decreasing.

This activity foreshadows the “tail of a series” (whether a series converges depends only on the tails).

Common point of confusion: some students will think that $a_{n}= 1/n$ provides a counterexample to 2. They are thinking that it is possible for $\displaystyle \int_{0}^{\infty}f(x) dx$ to be divergent while $\displaystyle \int_{7}^{\infty}f(x) dx$ is convergent as long as $\displaystyle \int_{0}^{7}f(x) dx$ is divergent.

Answers: 1, 2 are true. 3 is false.

This is one of the most important results about series, and one of the most commonly misused in tests and assignments. I would advise using this activity even more than once.

2, 3 are true. 1,4 are false.

These questions are hard. Most students try to do them “by feeling”, just by looking at them and guessing, without writing anything down, or thinking of definitions. That is pretty much impossible.

If I use this question, I aggressively tell students to rewrite everything in terms of partial sums. Otherwise, they will get nowhere.

For 1 and 2, we need to notice that $\displaystyle \sum_{n=k}^{\infty}a_{n}\; = \; S - S_{k-1}$. This is obvious to you and me, but students do not think of it easily.

The notation in 3 and 4 will confuse students. It may be worth it to write the first few terms of the series to make sure they understand what they are working with.

Answers:

• 1 and 2 are true.

• 3 is true. $\displaystyle \sum_{n=1}^{k}a_{n}= \sum_{n=1}^{\lfloor k/2 \rfloor}a_{2n}+ \sum_{n=1}^{\lceil k/2 \rceil }a_{2n+1}$, then use limit laws.

• 4 is false. Counterexample: $\displaystyle 1 - 1 + \frac{1}{2}- \frac{1}{2}+ \frac{1}{3}- \frac{1}{3}+ \frac{1}{4}- \frac{1}{4}+ \ldots$

In Video 13.6 students learn that series are linear. The video proves additivity, but leaves scalar multiplication (the proof in this activity) as an exercise.

The goals of this activity include

• We cannot assume all properties of finite sums carry on to infinite sums. We need to verify which ones do.

• To prove they do carry, just use the definition of a series as a limit of partial sums.

This proof will be accessible to students who watched the videos.

I use this activity early in Unit 13 to keep these results (which belong to Unit 12) fresh and relevant in students’ minds.

We will need to recall them when we get to integral test and comparison tests for series.

Answers: 1, 4 are convergent. 2, 3, 5, 6 are divergent.

This activity summarizes two essential families of series. We want students to be able to answer this in their sleep.

Answers:

Practice with standard applications of integral test and comparison tests.

• Slide 1 contains easier questions. We want everybody to eventually be able to solve these in their sleep. Students will be slower than you think.

• Slide 2 contains harder examples including some very challenging ones. You could easily spend a whole class just on this slide.

  When I use it, I tell students to focus on Question X, Y, Z (the ones we will actually discuss), but the rest are there to keep faster students busy.

Students need the practice, and they will take a lot of time. It is time well spent.

Answers:

• Slide 1. Use LCT with $\displaystyle \sum_{n}^{\infty}\frac{1}{n^{p}}$ for an appropriate value of $p$.

• Slide 2

  • Convergent: LCT with a geometric series

  • Convergent: $\displaystyle \ln n \ll n^{1/40}$ and use comparison tests

  • Convergent: LCT with $\displaystyle \sum_{n}\frac{1}{n^{2}}$

  • Convergent: Integral test.

  • Divergent: Integral test

  • Convergent: $\displaystyle e^{-n^2}< e^{-n}$, then integral test. Or $\displaystyle e^{-n^2}\ll \frac{1}{n^{2}}$.

This is one of the most important questions in the unit, because it summarizes many ideas. If a student understands everything, then it is delightfully simple. Otherwise, it is impossible.

However this does not mean you should necessarily use it in class! If students are not comfortable with the basics, you will be wasting a great activity and they will not be able to to solve it. If you do not have time for it, do not worry: it is included in the practice problems and students will attempt it when studying for the test.

Answers. Since $\displaystyle \sum_{n}a_{n}$ is convergent, we conclude $\displaystyle a_{n}\longrightarrow 0$.

• Convergent. Use LCT as $\displaystyle \lim_{n \to \infty}\frac{\sin a_{n}}{a_{n}}= 1$.

• Divergent. $\displaystyle \lim_{n \to \infty}\cos a_{n}= 1$.

• No conclusion. Examples $\displaystyle a_{n}= \frac{1}{n^{2}}$ and $\displaystyle a_{n}= \frac{1}{n^{4}}$.

• Convergent. $0 < a_{n}<1$ for large values of $n$. Therefore ${a_n}^{2}< a_{n}$. Use BCT.

Answer: It is always convergent – compare it with $\displaystyle 0.9999....$ using BCT.

This question is hard to use well. Students easily miss the point and do not understand what we are doing or why. Those who get it find the question easy (but perhaps satisfying). Those who don’t just wait without doing anything.

Simple question to check the statement of the alternating series test with very easy examples.

Answers: 2, 4, 5 are true; 1, 3, 6 are false.

These questions address a step in the proof of the Alternating Series Test (Video 13.13) that students may miss.

Answer: In general, the sequence $\displaystyle \{S_{n}\}_{n=0}^{\infty}$ is convergent iff the two sequences $\displaystyle \{S_{2n}\}_{n=0}^{\infty}$ and $\displaystyle \{S_{2n+1}\}_{n=0}^{\infty}$ are convergent to the same limit.

3 is true; 1, 2 are false.

The terms of these series only satisfy the hypotheses of the Alternating Series test eventually, but that is enough. That is the point of the question.

Some students mistakenly believe that we do not need to check the sequence $\displaystyle \left\{ \frac{n - \pi}{e^{n}}\right\}_{n}^{\infty}$ to be (eventually) decreasing. They mistakenly reason: it is (eventually) positive and convergent to 0, so it must be (eventually) decreasing.

A simple way to check monotonicity is to extend to a function and take the derivative.

Standard application of the second half of the Alternating Series Theorem.

Answer: The smallest partial sum that works is   $\displaystyle S_{2}= 1 - \frac{1}{3!}+ \frac{1}{5!}= \frac{101}{120}$   because   $\displaystyle \frac{1}{7!}< 0.001$.

Common error: Many students correctly calculate that $\displaystyle \frac{1}{7!}< 0.001$ and therefore think we need to use $\displaystyle S_{3}= 1 - \frac{1}{3!}+ \frac{1}{5!}- \frac{1}{7!}$ as an estimate.

This error comes from memorizing the formula without thinking about why it is true.

These two series have both positive and negative terms, but they are not alternating as in the Alternating Series Test. They are unlike any example students my have seen. The goal is to challenge them to be creative and come up with new tricks.

You will need to convince students to play and experiment. Most students reaction to this activity is “I do not know what to do. I will just wait for the answer”. Collaboration is essential for this activity. Making the goal explicit may help.

I like to use this question at the end of class, give them a bit of time, and not solve it.

Answers:

• $A$ is convergent. Group every two terms together and you have a new series that does satisfy the hypotheses of the Alternating Series test. Equivalently, think of the sequence of even partial sums.

• $B$ is divergent. Group every 5 terms together. Equivalently look at the sequence of every 5th partial sum.

  $$B \; \geq \; 1 + \frac{1}{6}+ \frac{1}{11}+ \frac{1}{16}+ \ldots$$

Small hint: there can’t be a counterexample to the Alternating Series Test, so which of the hypotheses is this series failing?

Many students realize that we need $\displaystyle \{b_{n}\}_{n}$ not to be eventually decreasing. Unfortunately, this by itself is not enough and most students get stuck.

Big hint: You can define $\displaystyle \{b_{2n}\}$ and $\displaystyle \{b_{2n-1}\}$ independently.

Students will figure it out with this hint, but unfortunately, it gives a little bit too much away. It would be better to guide to discover this hint themselves, but I do not know how.

Sample answer: $\displaystyle b_{2n}= \frac{1}{n}$ and $\displaystyle b_{2n-1}= \frac{1}{n^{2}}$.

Quick check of the definition of absolute and conditional convergence. The examples are easy if we remember the definitions.

Answer:

• is conditionally convergent

• is absolutely convergent

• is divergent. (Yes, the wording is misleading on purpose, just to check if they are paying attention.)

Questions 3 and 4 are the “absolute convergence test”. This is an essential result that students must know.

Questions 1 and 2, by contrast, are new questions. Students must think of them on the spot. Having all four questions together will confuse some of them.

Answers: 1 and 4 are true; 2 and 3 are false.

These two activities work fine together or independently (either one of them).

The goal is to make students think about the ideas behind the absolute convergence test, and the difference between absolute and conditional convergence (including the reason for the name).

Answers:

This activity works well as a warm up on the last session of Unit 13, before we go on practicing the Ratio Test.

This is NOT about the Ratio Test. Rather, it is a summary of all the important examples of series whose convergence is very easy to calculate. We want students to be able to answer these question in their sleep.

Answers:

1. Divergent (geometric)
2. Convergent (geometric)
3. Convergent ($p$-series)
4. Divergent ($p$-series)
5. Convergent (alternating series test)
6. Divergent (necessary condition)
7. Divergent (LCT)
8. Convergent (LCT)

Standard practice with the Ratio Test.

What to expect:

• Question 1 (convergent) is easy, but students are likely much slower than you expect.

• In Question 2 (divergent), some students have trouble simplifying $\displaystyle \frac{(2(n+1))!}{(2n)!}$

• In Question 3 (convergent), students will have trouble computing the limit.

• In Question 4 (divergent) the Ratio Test is inconclusive. This is on purpose, to remind students this may happen. Use BCT instead.

In Video 13.18 students learn the Ratio Test and a heuristic argument (not a formal proof) of why the theorem is true. The videos do not include the Root Test at all.

The goal of this question is to guide students to come up with a new theorem (intuitively; without necessarily writing a formal proof). It is a bit of a gamble: it may pay off and be very satisfying.... or it may flop and lead nowhere.

To clarify: students do not need to learn the Root Test in this course. In this activity they are practicing other skills; the goal is not to memorize the theorem itself.