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7h
comment Why does $\sum_{k=1}^{\infty}\sum_{\ell=0}^{k-1} = \sum_{\ell=0}^{\infty}\sum_{k=\ell+1}^{\infty}$
Why would it be the same? Some series are not absolutely convergent, only conditionally convergent. Where is the term $A_{k,\ell}$ you sum?
16h
comment All are sides of obtuse triangles
@RoryDaulton I used brute force, a computer program. I ran through a lot of integer quadruples $(a,b,c,d)$ where (WLOG) $0<a\le b\le c\le d$. I checked a bunch of inequalities to make sure all four combinations were in fact triangles, and to check that those triangles were in fact obtuse. Upon thinking a bit more, when $0<a\le b\le c\le d$ is known, we need to check only $a+b>d$ to make sure all combinations are triangles (triangle inequality). And we need to check only $b^2+c^2<d^2$ and $a^2+b^2<c^2$ to make sure every triangle is obtuse (from law of cosines, generalizing Pythagoras).
19h
comment All are sides of obtuse triangles
The smallest possible quadruple is: $6,6,9,11$. The first case where all four numbers are distinct (and hence no two of the four obtuse triangles are congruent): $7,8,11,14$
20h
comment All are sides of obtuse triangles
I guess this works for five positive real numbers as well.
20h
comment How do squares of non-right triangles relate?
Often opposite side-angle pairs are given "related" symbols, for example if one side/angle is called $A$, then the opposite angle/side is called $\alpha$, $a$ or similar (example: Wikipedia, Solution of triangles). However, you did not follow that practice (in your formula, $\alpha$ and $A$ must be adjacent). Just saying, so that people who are used to such a convention, will not get confused.
21h
comment Are there an infinite number of prime numbers where removing any number of digits leaves a prime?
Your sequence is a subsequence of OEIS A068669 (where they have the weaker requirement that all substrings (i.e. consecutive strings of figures) are non-composite). A068669 has twenty-four members.
21h
comment Are there an infinite number of prime numbers where removing any number of digits leaves a prime?
None of the three-digit examples survives if we decide to disallow the number $1$ as a prime number.
Apr
26
comment Are we allowed to compare infinities?
What is called limiting density here also has some other names. Wikipedia: natural density also notes the names asymptotic density and arithmetic density.
Apr
25
comment Maximum and minimum of of $f(x)=|x-1|+|x-2|+|x-3|$
Do you need to find "the" maximum only, or are you required to find all local maxima? You should be able to realize that the graph of $f$ has a shape like a W.
Apr
24
comment Why is Euler's number used as a base for logarithms?
Of course this is the same property of $e$, but the only solutions $y=ca^x$ to the simple differential equation $\frac{dy}{dx}=y$ are those which have base $a=e$.
Apr
24
comment Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational.
@BillDubuque Yes. This proof seems to use the property that when elements $p$ and $q$ have no common divisor (other than units), then $p^2$ and $q^2$ have no common divisor (other than units). It sounds like a property that holds only in some rings (which ones?).
Apr
24
comment Find the point where this function is not locally invertible
My example from the first comment where $\phi(x)=x^2$ is not invertible, but still has $\psi(x)=\sqrt{x}$ (I chose the sign arbitrarily). It seems that I should try with $\phi:\mathbb{R} \to \mathbb{R}_{\ge 0}$ and then $\psi: \mathbb{R}_{\ge 0} \to \mathbb{R}$. Then $\phi\circ\psi$ (first take square root, then square) is really the identity, on $\mathbb{R}_{\ge 0}$. But the other composition, $\psi\circ\phi$ (first square, then take square root) is not the identity on the entire set $\mathbb{R}$. Actually $\psi\circ\phi$ is the absolute value function. Isn't this analogous to your example?
Apr
23
comment Find the point where this function is not locally invertible
Did you check both $f\circ g$ and $g\circ f$? Also, what set should be the domain of $g$ (the range of $f$)?
Apr
23
comment Find the point where this function is not locally invertible
(edited) Your function $g$ contains some plus/minus ($\pm$) signs in its definition. Does $g$ really assign just one point to each point in its domain? Is it a function? Addition: Maybe this is like saying that for the non-invertible function $\phi(x)=x^2$ you have still found and inverse function $\psi(x)=\pm\sqrt{x}$.
Apr
23
comment Is there an integral that proves that $\sin \tan 1\lt 1$?
If anyone does not have a calculator, the approximate values are $$1 - \sin\tan 1 = 0.0000896$$ and $$\frac{\pi}{2} - \tan 1 = 0.0134$$
Apr
23
comment rational function cancellation
Strictly speaking, $x \mapsto \frac{x(x-1)}{(x-1)}$ does not determine a mapping from $\mathbb{R}$ to $\mathbb{R}$. If you have a mapping from $\mathbb{R} \setminus \{ 1 \}$ to $\mathbb{R}$ it is OK to simplify, when you remember that the mapping is from $\mathbb{R} \setminus \{ 1 \}$ to $\mathbb{R}$, also after the simplification.
Apr
21
comment Why do we use degrees?
Evidently, $2\pi$ has a factor $2$ which is a whole number (joke; use $\tau$, however).
Apr
17
comment How do you create an alternating series with the sign being the same twice in a row?
Well, if we group every second term we get stuff like $\sum_{m=1}^\infty (-1)^{m-1} \left( \frac{1}{2m-1} \pm \frac{1}{2m} \right)$ which can be handled by Wolfram Alpha, so the answer is yes.
Apr
16
comment How do you create an alternating series with the sign being the same twice in a row?
Suppose I take some simple series and use these signs with them, do I get something well-known? I was thinking $$\sum_{n=1}^\infty \frac{(-1)^{n(n-1)/2}}{n}$$ i.e. the harmonic series taken with signs like this, is the sum of this series known?
Apr
12
comment How many points to prove a trigonometric identity?
Would you choose only numbers where you could evaluate the expression exactly, or would you use approximate floating-point calculations?