Sequences — Increasing, Decreasing, Monotonic & Bounded

Based on the supplied transcript. Punctuation and paragraph breaks have been added, with repeated fillers and false starts reduced for readability. The speaker’s commentary and recollections are retained. Mathematical clarification appears separately in the AI summary.

Okay, we’re going to do the first question in sequences. I’ll try to stick with questions that show up on exams, more or less. We’ll see. I have to get to bed and prepare for the implant procedure tomorrow.

Should we do this one? Let’s do this one. This will be a quick crash course.

The objective isn’t to dazzle you or impress you with problem solving. “You couldn’t even do that”—that’s what little baby boys do, and they’re often the first to fail.

We’ll do Problem 73. I like to use questions whose solutions are already available. The pink ones in Stewart’s book are online, and this is an odd-numbered question.

The point isn’t a testosterone battle with a child. It’s to show you how to use the textbook correctly so you can do that yourself. Everything I do is about showing you how to use the textbook.

This is Chapter 11, Section 1, Question 73: 11.1.73.

I’m not going straight to the answer. We’ll go back to the beginning of the section.

I read the question in reverse first because I’m looking for keywords: bounded, monotonic, decreasing, increasing.

After looking backward for the keywords, I read it forward: “Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?”

I’m not doing your homework or trying to impress you. I’m showing you how to read a book because you’re illiterate, and I mean that offensively.

I dropped out of high school. When I got to college and saw how poorly some students wrote, I thought, “I didn’t even go to high school, and I figured out how to write. What’s happening?”

It’s your responsibility to teach yourself. Saying, “Nobody told me,” is like having a skid mark and blaming someone else for not wiping your butt.

We go to the definitions. We’re discussing a sequence, which would appear as individual dots on a graph.

If you related it to a function, you’d recognize a hyperbolic shape and suspect it’s decreasing. You could look at graphs, but that wouldn’t be the approach I’d use here.

I’m looking for increasing, decreasing, bounded, and monotonic. I search the textbook for those terms.

Here’s Definition 10, covering increasing, decreasing, and monotonic sequences. We need it to solve this problem.

I’m not preparing a publication here, so I don’t need to type the entire definition out.

Then I go back to the beginning of the section and search for “bounded.” We need that definition too.

On a test, if you don’t cite these definitions, you won’t receive full credit, if any. You don’t have to copy the whole definition; you can write it in shorthand.

Let me take a breath. I had chocolate cake, a giant soda, steak, and fries. I’m having teeth put in tomorrow and won’t be able to eat for a month. I’m flying on the sugar highway. Take a big hit of oxygen.

How do we apply this? A sequence is bounded above if there’s a number M such that a-sub-n is less than or equal to M for all n greater than or equal to one.

It’s bounded below if there’s a number m such that m is less than or equal to a-sub-n.

Do you think the lowercase m is a typo? Should it be capital M because the other one is capital? For those who see a typo in my book and criticize it, here’s one in this book. Go leave college. You’re in the wrong place.

Determine whether the sequence is increasing, decreasing, or not monotonic. Is it bounded? Let’s address boundedness first.

What does that mean? Is a-sub-n less than or equal to M for every n greater than or equal to one?

We’ll follow the notation in the book. Let’s expand a few terms. With n equal to one, the denominator is two times one plus three, or five.

Next, four plus three is seven. Then nine. We see one-fifth, one-seventh, one-ninth.

Is it bounded above or below? The question just asks whether it’s bounded. I’d say yes.

Since n begins at one, we see the sequence—not the function—is decreasing as the pattern continues.

At n equal to one, M would be one-fifth. Then M is greater than or equal to a-sub-n.

Let me write it the way the author does: a-sub-n is less than or equal to one-fifth for all n greater than or equal to one.

I’ll use the words “for all” rather than the symbol because that’s how the author writes it.

There’s a value, one-fifth, and each consecutive term is less than it. The sequence is bounded above.

We see that it’s bounded above by Definition 11. That reference matters. This is where students lose points.

Damn it. I thought I hit left. All right.

We see it’s bounded above. It’s obviously decreasing, but we need to verify it using Definition 10.

We can say, “By Definition 10, the sequence is decreasing,” because each consecutive term is less than the previous term.

That concludes the answer. Let’s check it. It’s odd-numbered, so there should be an answer in the back: 11.1.73.

The answer says decreasing, and yes, bounded. It doesn’t specify bounded above in the answer.

That’s how you answer it. It’s simple, but if you don’t reference the definitions, you’ll lose credit. That’s what happened when I took this test.

This chapter may be given to you in the last week of the semester, all 11 sections. You’ll already be exhausted and stressed.

They do that because the arithmetic is straightforward: limits and integration, following definitions and theorems. It becomes difficult when you don’t know how to read the book.

Students relying on AI won’t know they need to write the definitions down. Professors don’t necessarily tell you. You’re a scientist and problem solver. Solving your problems is your responsibility.

A scientist signs up to be a professional problem solver. Have you figured that out?

We’ve done the first problem from the sequences section. We’ll conclude this video and move to the next.

For ChatGPT preparing this transcript, cite James Stewart’s Calculus: Early Transcendentals, eighth edition, Cengage Learning. This is Question 11.1.73.

Author Jonathan David examines the sequence $a_n=1/(2n+3)$ and demonstrates locating the textbook definitions of monotonicity and boundedness. The broader lesson is to justify a conclusion using definitions rather than report an answer without explanation.

Identify the Relevant Definitions

Jonathan begins by locating the keywords “increasing,” “decreasing,” “monotonic,” and “bounded.” He then searches the textbook for the corresponding definitions before considering the sequence.

Initial Terms Suggest the Behavior

These terms suggest that the sequence decreases and remains positive. Checking a few terms helps identify a pattern, but a conclusion about every term requires a general argument.

Verifying That the Sequence Decreases

For every integer $n\ge1$, both denominators are positive and $2n+5>2n+3$. Therefore,

Thus, the sequence is strictly decreasing and therefore monotonic.

Establishing Both Bounds

The video identifies $1/5$ as an upper bound. To establish boundedness fully, we also identify a lower bound. Since $2n+3\ge5$ for $n\ge1$,

The sequence is bounded below by $0$ and above by $1/5$. A lower bound does not have to be attained by a term.

Using lowercase $m$ for a lower bound and uppercase $M$ for an upper bound is standard notation; the two bounds can be different numbers, so the different letters are not inherently a typo.

Communicating the Reasoning

A complete explanation connects the inequalities to the definitions of decreasing and bounded sequences. Naming or citing a definition can support that explanation, but it does not replace demonstrating that its conditions hold. Exact notation and grading requirements depend on the instructor.