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seen Jan 19 at 21:37

Jan
2
comment An Additional Rule for Calculus
Your counterexample isn’t, since by definition $x^x=\exp(x\ln(x))$. You might need something like $_2F_1(1,x;1/2;x)$, although it feels like there should be simpler examples.
Jan
2
comment Show that $\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $ using Birkhoff Ergodic Theorem
I assume the product goes to $n$, not $\infty$. As written, that limit exists if and only if the infinite product converges. In which case the limit is 0 or 1.
Dec
31
awarded  Good Answer
Dec
30
revised Why do bases of infinite dimensional spaces need to be orthonormal?
a_n must be inside the sum, since it depends on n
Dec
30
suggested approved edit on Why do bases of infinite dimensional spaces need to be orthonormal?
Dec
29
comment Chicken Problem from Terry Tao's blog
Nice. But why must $u$, $v$, and $w$ be pairwise different?
Dec
29
comment What parts of a pure mathematics undergraduate curriculum have been discovered since 1964?
We did Pollard’s rho method in second year (Germany), in a course focused on getting the computer do advanced computations. (Kind of a math-centered programming course, if you will, but still a math course, including the theoretical discussion. No complexity analysis or anything like that, though.)
Dec
28
comment Can we teach calculus without reals?
While I’m a fan of differential algebra (which is the term I learned for what you described), it’s probably not a substitute for calculus in a school setting. You really want things like the mean value theorem, and without reals (or, if you take a rather complicated route, constructible reals), it’s just plain wrong.
Dec
28
comment Can we teach calculus without reals?
I believe what you’re getting at here is what is known as “constructible reals.” They are, of course, countable. For a proper definition, though, you need to specify a language (in the CS sense) for describing the numbers, and in my experience, they are not really easy to work with.
Dec
26
comment Why do we need to assume continuity in the proof of the chain-rule?
@Nick I’m not sure I see how it fails. What’s written there is pretty much the definition of being continuous at $x$, no continuity anywhere else is needed.
Dec
26
answered Finding Tangent Line to Graph
Dec
26
awarded  Nice Answer
Dec
23
comment Easy example why complex numbers are cool
Most? I’d say most integrals cannot be solved in closed form, but we may simply be looking at different distributions over the space of possible integrands…
Dec
23
answered Easy example why complex numbers are cool
Dec
18
comment How to determine if the sequence $a_n= (30+12\arctan(n!))/6^n$ is divergent or convergent
Wasn't the question about the sequence itself, not the series?
Nov
6
revised An element of a group has the same order as its inverse
an exponent was missing its ^. Took out a few {} because I need 6 characters
Nov
6
suggested approved edit on An element of a group has the same order as its inverse
Oct
25
comment Why doesn't $\arccos x = -\tfrac12\sqrt{3}$ have any solutions?
No need to limit yourself to real $x$. There's no complex $x$ with that property either. (In both real and complex numbers, that is true if you take the standard primary branch.)
Jun
5
comment 'Obvious' theorems that are actually false
@Asaf, maybe you could clarify what exactly you mean by “height” here. Each element of the set you defined clearly has order $\omega$, and while I believe we mean the same thing for “the $n$-the level” when $n$ is finite, if I read $f\upharpoonright n$ as $f$ limited to the preimage set $\{0,\dots,n\}$. Now, none of these actually is in your set. Each element of your set is the union of a countable number of such finite sequences, which you can choose to view as finite paths. But what would, with your definition of “height,” be such a tree of height $\omega$?
Jun
5
comment 'Obvious' theorems that are actually false
@Asaf, I fail to see how you get order type $\omega+1$ entries in elements of the set $\{0,\dots,9\}^\mathbb{N}$. E.g., I think we agree there is exactly one element in this set corresponding to the decimal expansion of $\pi-3$. If that element, as you say, has order type $\omega+1$, then it has a last digit. Which one is that and why? And how does it come to be in a function from $\mathbb{N}$ to $\{0,\dots,9\}$?