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I currently work as a substitute teacher at a local high school and the topic in one of the classes is sequences, series and limits. Because I always disliked learning about a topic without having some sense of what problem it addresses or what new possibilities it opens, I wanted to present a motivating example in the first lesson. What I had in mind was an application of sequences, series or limits that is easy enough to grasp without already knowing the topic but still reliant on its ideas to motivate its study.

To clarify some of the terms I am using: By sequences, series and limits I mean the basic theory of sequences (explicit formulas, recursive formulas, arithmetic sequence, geometric sequence, divergent and convergent sequences), the simple series (arithmetic series, geometric series, divergence of the harmonic series) and the definition of convergence using an epsilon bound.

After doing some research on the web I went with Zenos paradox of Achilles and the Tortoise and after presenting it, letting the students discuss it among themselves and arguing Zenos position, I showed them how the paradox can be resolved with an infinite series. In the last part however, I had to simply state that this infinite summation yields a finite value and that proving this fact and similar things would be the content of the following lectures. Afterwards I felt like I had presented a solution that wasn't really understandable and that left people rather disappointed than hungry for more. So my question is if any of you know motivating examples for the study of sequences, series and limits to get high school students interested in the topic? Ideally I would like these examples to highlight the importance of dealing with infinities. I know there are plenty of cool problems involving sequences and series that connect to geometry, physics or finance ( for example here: Examples of arithmetic and geometric sequences and series in daily life ) but I haven't found ones that met my criteria of demonstrating the need for sequences, series and dealing with infinity in an accessible way.

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    $\begingroup$ Off the top of my head, as I don't have time to write a decent answer now -- Every infinite decimal expansion of a real number is an infinite series, and those that arise from finding (by hand) the (eventually periodic) expansions associated with rational numbers can be reversed back into "quotient of integers form" using geometric series. Problems like this and this and this can be solved using infinite geometric series. $\endgroup$ Aug 21 at 15:57
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This application is known as "gross-up" in accounting.

You run the finances for a small business. The boss would like to give an employee a \$100 bonus for their hard work. However, the employee's earnings are taxed at 20%. The boss says "okay fine; pay them \$120 and then they will get \$100 out of it."

a) If you pay the employee \$120, how much money do they actually take home?

b) Can you cover the difference by adding a few dollars to the payment?

c) How much money should you actually pay the employee for them to take home \$100?

The final answer of \$125 can be found by either

  • $(1-0.2)x = 100$

or

  • $x = 100 + 20 + 4 + 0.80 + \ldots$.
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    $\begingroup$ Funny, I just finished tutoring a College freshman taking her math class this summer. And explained a similar problem, but with a tip. The credit card bill shows $120, and you know you tipped 20%. How much was the tip alone? Not 20% of 120, of course.... We push kids to calculus, but I worry about how many haven't mastered percents by the time they graduate HS. $\endgroup$ Aug 22 at 12:34
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I'm not sure that starting with an applied motivation (derivation, word problem) is the best way to introduce this topic. Look at how your experiment failed. This is because "word problems are hard". Really, they are. One of my big criticisms of most PDE texts is introducing the topic as an engineering problem. And I say this as an applied guy!

Had an ODE teacher who always introduced new techniques with a motivating applied problem. But I found this much harder than learning the technique first. Then, perhaps practicing it later, after mastery, on some applied problems (probably well after sufficient drill on simple non-applied exercises). Had a sub for that course, who just stressed for us the main techniques and exercises and the like that were "just algebra" and it was really refreshing for a class that was getting annoying to me.

Now, this doesn't mean zero motivation. But I think it's really sufficient to just mention that these are techniques useful in such and such fields very quickly and without proving how or deriving it. Like if I were teaching imaginary numbers the first time, I might mention that they're useful in electronics. But I'm not getting into ELI the ICE monster. It's just a throwaway remark to say that the topic has uses.

This doesn't mean you shouldn't know the applications. Yes, you should, because you should have a deeper knowledge of the topic than your students. It enriches you. But you don't need to show that. Remember these topics are new to the students. Every fresh crop of kids brings fresh ignorance. So, try to introduce things gently and progressively.

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There are lots of good physics examples involving equilibrium. For example, you can set up a pendulum and show how the amplitude forms a sequence that decays exponentially toward zero, or describe putting a hot object and a cold object in thermal contact and measuring the temperature at some fixed time interval.

When you ask students questions like, "How could we give a numerical description of how strongly friction affects the pendulum?," they will often come up with answers like "Count how many swings it takes until it stops moving." They usually find it pretty intellectually interesting to consider the possibility that there is never a discrete point at which the motion stops (although eventually the motion will certainly become too weak to be detectable given background perturbations such as air currents and vibration from passing trucks).

Epidemiological or biological models might provide a motivation similar to Zeno's paradox but without the silly philosophical cruft. When you consider the population of a certain species over time, there can be cases of positive feedback, which cause exponential growth or decay, and also negative feedback, which causes oscillations. The exponential decay cases are mathematically the same as Zeno's paradox. E.g., we expect the number of people who have been exposed to covid to exponentially approach the entire population. To make this a discrete sum, simply pose the model as a discrete one, by talking about the total cases per year or something.

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The common puzzle of giving a few terms and asking for the next are examples of (generating) sequences by some particular rule.

A series is just a sequence, summed together. Ask e.g. for the sum $1 + 2 + \dotsb + n$ (a finite series).

For infinite series/limits, ask for $0.1 + 0.01 + \dotsb$, you can plot the sucsesive terms and show they get ever nearer $1 / 9$. Or use $1 + 1/2 + 1/4 + \dotsm$. Bonus point is to point out that e.g $0.1 + 0.1 + \dotsm$ doesn't have a limit, so that the terms tend to zero is a requisite for convergence, then show that $1 + 1/2 + 1/3 + \dotsm$ doesn't converge either (simplest proof is to compare with $1 + (1/2) + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + \dotsm$ and note that each term is less than the respective term of the series and each parentesis adds up to $1/2$).

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    $\begingroup$ I am Happy to sede a familiar user also here.😃😃😃😃+1 $\endgroup$
    – Sebastiano
    Sep 29 at 21:47
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Taylor series are really useful tools for solving a number of calc 2 problems without analytic solutions. It's a bit more advanced than some of these other examples, but might still be viable as a way of tying a number of topics together.

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