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 ...
- 581
15
votes
Accepted
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.
- 24.4k
11
votes
Accepted
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 ...
- 21.2k
10
votes
Accepted
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 ...
- 9,079
10
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 ...
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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 ...
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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? (...
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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 ...
- 10.7k
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 ...
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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 ...
- 2,484
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 ...
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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 $n \ln n$ (...
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5
votes
Accepted
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 $\frac{p}{q}$ ...
- 10.3k
5
votes
How can I teach my students the difference between a sequence and a series?
Sequence is just a function of the type $f:\mathbb{N} \to \mathbb{R}$. It is common to list the elements of this sequence as $$(a_1,a_2,a_3,\ldots,a_n)\,.$$ One example is the sequence of all even ...
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5
votes
How can I teach my students the difference between a sequence and a series?
Well, the first step to recovery is admitting that you have a problem. Just by acknowledging that students have this misconception, you're already well on your way to resolving it. Students are a ...
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 ...
- 1,227
5
votes
Accepted
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
$$
\sum_1^\infty \frac{n+1}{n^4+n+7}
$$
...
- 24.4k
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 ...
- 483
5
votes
Accepted
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 ...
- 1,093
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 ...
- 3,258
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 + ...
- 12.3k
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 ...
- 3,258
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 ...
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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 ...
- 10.7k
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 ...
- 31
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 ...
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3
votes
How can we explain intuitively the convergence and divergence of these two series?
Intuitively, to me, it means that if you take the positive number line, put a blue dot at every prime, and a red dot on all the the numbers of the form $n^{1.000000000001}$, then eventually, very far ...
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2
votes
For calculus students, what should be the intuition or motivation behind series?
The fact of the matter is series is sort of viscerally off the beaten track of limits/derivatives/minmax apps/integral toolbag/integral apps. It's there because it is needed later (and covered more) ...
- 21
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