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17 votes

Examples of arithmetic and geometric sequences and series in daily life

I tutored a student who came with a kind of problem I had never seen before and found quite refreshing. It was something like: A child is being pushed on a swing by their father, reaching a maximum ...
pjs36's user avatar
  • 581
15 votes
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What strategy for picking convergence tests for series do you teach?

I made this flowchart for my students last time I taught this stuff. Not the best visually, but I think it effectively conveys my thought process.
Steven Gubkin's user avatar
11 votes
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Motivating example for sequences, sums and limits in high school

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 ...
Chris Cunningham's user avatar
11 votes

How can we explain intuitively the convergence and divergence of these two series?

Look at a simpler example first: $(1.000000000001)^n$ compared to $0.9999999999^n$. Do they accept that the first sequence tends to $\infty$ and the second to $0$ even though it would take quite a ...
KCd's user avatar
  • 3,536
10 votes
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Examples of arithmetic and geometric sequences and series in daily life

Here are a few more examples: the amount on your savings account ; the amount of money in your piggy bank if you deposit the same amount each week (a bank account with regular deposits leads you to ...
Benoît Kloeckner's user avatar
9 votes

Examples of arithmetic and geometric sequences and series in daily life

I like to explain why arithmetic and geometric progressions are so ubiquitous. Using the examples other people have given. Geometric progressions happen whenever each agent of a system acts ...
Anita's user avatar
  • 91
6 votes

Examples of arithmetic and geometric sequences and series in daily life

This is a nice demonstration that $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\cdots = 2 \;.$$                     (Image from Wikipedia ...
Joseph O'Rourke's user avatar
6 votes

Comparison Tests in Calculus

You should do option (2). As you already mentioned, there are series that cannot be done through limit comparison and can only be done with direct comparison. Why would you purposely leave those out? (...
Aeryk's user avatar
  • 8,019
6 votes

Comparison Tests in Calculus

I believe option #2 (teach both and methods for choosing) is best. I'll try to illustrate with examples. $\displaystyle{\sum_{n=1}^\infty \frac{n}{n^2+1}}$ The intuition (which most students see ...
Brendan W. Sullivan's user avatar
6 votes

Motivating example for sequences, sums and limits in high school

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 ...
guest troll's user avatar
6 votes
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Intuition explanation about Lebesgue measure zero of the rational numbers

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ ...
user52817's user avatar
  • 11k
6 votes

What strategy for picking convergence tests for series do you teach?

I think this is one of those places where teaching a detailed strategy is a form of "teaching to the test" that is counterproductive for the students' intellectual development. It's ...
Alexander Woo's user avatar
6 votes

How can we explain intuitively the convergence and divergence of these two series?

For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width). The $n$th prime is asymptotically $n \ln n$ (...
Justin Skycak's user avatar
5 votes
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Finite sum of infinite series

"Infinite sum" is in common use, so it should be acceptable to say that $2$ is the value of the infinite sum $1+\frac{1}{2}+\frac{1}{4}+\ldots$. But students need to be very clear that the ...
Will Orrick's user avatar
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5 votes

For calculus students, what should be the intuition or motivation behind series?

I start on the other end of the spectrum. I start off by Noting that computers can't compute sine, cosine, or anything like it. Computers can basically only compute polynomials. Sine and cosine ...
johnnyb's user avatar
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5 votes
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Comparison Tests in Calculus

Another big advantage of direct comparison (if you have the time) is that it can be used to establish error estimates. For instance, if I want to estimate $$ \sum_1^\infty \frac{n+1}{n^4+n+7} $$ ...
Steven Gubkin's user avatar
5 votes

Examples of arithmetic and geometric sequences and series in daily life

They might be interested to know about both Moore's Law and "Nielsen's Law". You've probably heard about Moore's Law, where computer complexity doubles about every two and a half years. Internet ...
David Elm's user avatar
  • 485
4 votes

Comparison Tests in Calculus

While the harmonic series shows us that $a_n$ tending to $0$ is not sufficient to guarantee convergence, the comparison and limit comparison tests are strong "almost replacements": they justify the ...
KCd's user avatar
  • 3,536
4 votes

Generating function example

Why not give the generating function for the Catalan numbers as an example? For a quick overview: The Catalan numbers ($C_n$, A000108) is defined as the number of ways that parenthesis can be ...
CrSb0001's user avatar
  • 295
4 votes

Generating function example

I suggest using generating functions to find two dice whose sum has the same probability distribution as a pair of d6's. You can find this written up in the Sicherman Dice wikipage (where the section ...
Benjamin Dickman's user avatar
3 votes

Examples of arithmetic and geometric sequences and series in daily life

When I think of a geometric sequence, I think of something where the initial input value = 1, not 0. Most interest problems would start at time = 0, so I would exclude these unless you said something ...
Bryan Baz's user avatar
3 votes

Comparison Tests in Calculus

I tend to teach option (2.) for many of the reasons already listed in the existing answers. If we just teach (1.) then the problem solving becomes formulaic which means the whole endeavor is ...
James S. Cook's user avatar
3 votes

Motivating example for sequences, sums and limits in high school

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 ...
user17655's user avatar
3 votes

Motivating example for sequences, sums and limits in high school

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 + ...
vonbrand's user avatar
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3 votes

What strategy for picking convergence tests for series do you teach?

First, let's ignore geometric series and $p$-series because those are standard examples. For infinite series with positive terms, if you really understand how sequences grow then almost all examples ...
KCd's user avatar
  • 3,536
3 votes

Proof that convergent Taylor Series converge to actual value of function

If the power series $\sum_{j=0}^\infty a_j z^j$ converges to some function $f(z)$, then the Maclaurin series of $f(z)$ is $\sum_{j=0}^\infty a_j z^j$. But the converse need not hold. It could happen ...
Gerald Edgar's user avatar
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3 votes
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Generating function example

Chapter 6 of "Applied Combinatorics" by Alan Tucker contains several elementary but powerful examples and exercises of generating functions. Example 4 asks for the generating function of the ...
Mike Z.'s user avatar
  • 397
2 votes

Proof that convergent Taylor Series converge to actual value of function

Functions whose Taylor series converge to the original function are called analytic. How do you know if a function is analytic? For teaching basic calculus it probably suffices to know that $\exp,\cos,...
Michael Bächtold's user avatar

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