KCd
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Unique candidate that fails
19 votes

Theorem: The largest positive integer is $1$. Proof: If $n$ is a positive integer and $n \not= 1$ then $n^2 > n$, so there is an integer larger than $n$. Thus the largest integer has to be $1$. ...

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How can I convince students that Fourier series are useful?
13 votes

The Fourier coefficients contain useful information that is not at all apparent just from the shape of the periodic function itself. For example, when Fourier series are used for musical vibrations we ...

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Simple "real world" l'Hôpital's rule problem?
12 votes

In thermodynamics and information theory the function $-x\log x$ for $0 \le x \le 1$ occurs in the definition of entropy. Just Google "entropy" and see. But that formula is not really defined at $x = ...

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How does knowing more about mathematics help one's teaching of lower level course, such as calculus?
11 votes

The title of the question asks how knowing "more about math" can help in teaching calculus, while the question itself asks specifically about the math learned towards a masters degree or a doctorate. ...

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Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III?
8 votes

You can find a proof that goes back to Gauss, which is based only on multivariable calculus (double integrals and partial derivatives) at http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/...

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Why are we even studying cyclotomic polynomials?
Accepted answer
5 votes

Can we grant that the students think roots of unity are worthwhile? If so, point out that one of the ways to understand roots of polynomials (like roots of $x^n-1$) is to understand the lowest degree ...

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What is most motivating way to introduce Fermat's Little Theorem
5 votes

Look at patterns in decimal expansions: what is the period of the repeating decimal of $1/n$? From numerical data, the period is at most $n-1$, and you only get equality when $n = p$ is prime (but not ...

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Inability to work with an arbitrary mathematical object
5 votes

Can all of your students correctly explain what the supremum and infimum of a set are? If not, they will be unable to reason about these concepts. It would be like expecting a student to do something ...

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Are there any applications of $x^x$?
4 votes

Logarithmic differentiation is used in mathematics when dealing with infinite products in complex analysis. Note the construction of $f’/f$ from $f$ does not require logarithms of $f$ as a middle ...

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Why do we teach the Rational Root Theorem? (high school algebra 2)
4 votes

Theorem: For every integer $m$, the polynomial $x^3 - mx^2 - (m+1)x - 1$ is irreducible among polynomials with rational coefficients. Proof: This polynomial has degree $3$, so if it is a product of ...

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Interesting settings for exponential growth or decay
4 votes

One basic model of atmospheric pressure has it decaying exponentially as a function of height above sea level. Two places to look for this are https://people.clas.ufl.edu/kees/files/...

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Comparison Tests in Calculus
4 votes

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 ...

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How to catch students from different subjects' interest to math?
4 votes

I am not here to bring hopeful news, but a harsh dose of reality. There is no simple way to make most students grasp the $\varepsilon$-$\delta$ definition of a limit in a first calculus course and I ...

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What is the intuition behind the limit superior?
4 votes

The idea behind the limsup that you write is not simple and will not convey a concise intuition. The shortest description of the limsup of a sequence needs two steps: (1) The audience has to know ...

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

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 ...

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Should homework be graded in an undergraduate math course?
3 votes

You complain about access on the internet to solutions to textbook problems. Here is a way around that: write your own homework problems.

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Explaining the intuition for why finding roots of polynomials is hard
2 votes

We can rescale a polynomial so its leading coefficient is $1$ without changing the roots. So let's focus on polynomials with leading coefficient $1$. If its degree is $n$ then it has $n$ ...

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How to prepare for lecturing in a non-fluent foreign language?
2 votes

I have given lectures numerous times in Russian. The main hurdle at first was notation, since books never discuss this. Ahead of time I thought of a lot of the notation I would need (how to pronounce ...

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At what point in the curriculum should the tensor product be introduced?
2 votes

In a math department (not physics department) I'd introduce it after some more familiar constructions (quotient groups, cyclic groups with a choice of generator,...) have already been described a ...

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Direct applications and motivation of trig substitution for beginning calculus students
2 votes

Trigonometric substitution is related to how Newton first showed the derivative of $\sin x$ is $\cos x$. See https://hsm.stackexchange.com/questions/3174/how-were-derivatives-of-trigonometric-...

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How can I explain why we need proofs to someone who has no experience in mathematical thinking?
0 votes

Does your friend get bothered by multiplication? You say $+$ and $-$ are okay but division is problematic, so you skipped over multiplication. If your friend is okay with multiplication, and knows ...

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