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 $1$. ...

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 $x = ... View answer 11 votes The title of the question asks how knowing "more about math" can help in teaching calculus, while the question itself asks specifically about the math learned towards a masters degree or a doctorate. ... View answer 8 votes You can find a proof that goes back to Gauss, which is based only on multivariable calculus (double integrals and partial derivatives) at http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/... View answer Accepted answer 5 votes Can we grant that the students think roots of unity are worthwhile? If so, point out that one of the ways to understand roots of polynomials (like roots of$x^n-1$) is to understand the lowest degree ... View answer 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 ... View answer 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 ... View answer 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 ... View answer 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 ... View answer 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/... View answer 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 ... View answer 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 ... View answer 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 ... View answer 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 ... View answer 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. View answer 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$n$... View answer 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 ... View answer 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 ... View answer 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-... View answer 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 ...