Hendrik Vogt
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 Jan 21 awarded Nice Answer Oct 29 comment How much do we really care about Riemann integration compared to Lebesgue integration? @B.S.Thomson: I don't claim that my answer could be taken as a reason to teach the Riemann integral, and you're right that ergodic theory wouldn't really suffer. However, the proof of the theorem is straightforward if you work with the definition of the Riemann integral. I didn't think about how it would work with the characterization you give. Oct 28 comment How much do we really care about Riemann integration compared to Lebesgue integration? I'm a bit late to the answering party, but I hope my answer gives additional insight. Oct 28 answered How much do we really care about Riemann integration compared to Lebesgue integration? Oct 17 awarded Yearling Aug 28 comment How does one denote the set of all positive real numbers? @user254665: Well, certainly it means the positive reals, but now ask them what they mean by "positive" :-) Seriously, I know mathematicians who mean "$\ge0$" and other who mean "$>0$". May 1 comment What is cosine to the power of zero? @AlexR: I'd really like to see a reference to a book where they leave $0^0$ undefined. The reason for my comment was the following: I've been teaching calculus for years, and many students had learned in school that $0^0$ is undefined. Then suddenly we have power series and stuff like that where $0^0$ appears naturally, and one needs to know that it's equal to $1$. So call it my mission to bring the gospel $0^0=1$ to the world. May 1 comment What is cosine to the power of zero? @AlexR: In analysis, one always defines $0^0=1$, but you're right that one has to keep in mind that $(x,y)\mapsto x^y$ is not continuous at $(0,0)$. In set theory, $0^0=1$ since $0^0=\varnothing^\varnothing=\{\varnothing\}$. I'm not 100% sure about other areas of mathematics. Where does it remain undefined, usually? May 1 comment What is cosine to the power of zero? I don't understand why you write "undefined else": $0^0$ is defined as $1$. And that's important, how would you write power series without this definition? (As in $\exp(x) = \sum_{n=0}^\infty x^n/n!$) Oct 17 awarded Yearling Sep 19 comment Series that converge to $\pi$ quickly @Ekuurh: You're right, the formulation "can be used" might be a bit misleading. The formula can be used for what I claim, but with increasing $n$ the computing time increases like $n\ln n$. Oct 17 awarded Yearling May 28 comment Surprisingly elementary and direct proofs Can you please explain a bit about the part "the computation can be done directly without appeal to the general minimum modulus principle"? This interests me very much! (I didn't do that proof in my lecture because I didn't have the general minimum modulus principle available.) May 26 comment How do you respond to “I was always bad at math”? @Steve: Maybe it depends on what you mean by "hard" and "struggle". The first thing I remember for which I'd use those words is closure operators, and this was in my first year at university. But that doesn't mean that I understood everything before immediately at the first read. So I come to the same conclusion as Brian, and would only say that math is hard for most people. May 25 comment Infinite series and its upper and lower limit. @Brian: Ah, thanks, now I got it! I interpreted the fact as the positive outcome of the ratio test for the modified series, which induced me to post my first comment. Thanks again! May 25 comment Infinite series and its upper and lower limit. @Brian: Ah, I think now I get your point. The result I use is simple enough, but it has to be proved, and one has to say that one uses it. Agreed! However, I maintain that your formulation "you can’t make use of the fact to show that it converges" is not true. Or maybe I don't understand "can’t make use" well enough, I'm not a native speaker? I'd rather say something like "this fact alone doesn't yet show that it converges". May 25 comment Infinite series and its upper and lower limit. @Brian: OK, let me formulate it differently. The series has positive terms, so the sequence of its partial sums is increasing. And an increasing sequence is convergent if any of its subsequences is convergent. The OP proved convergence of one subsequence, so we're done. May 25 comment Infinite series and its upper and lower limit. @Brian: "you can’t make use of the fact to show that it converges" is not true. You should make use of that fact, and indeed you should do it properly. The series in question has positive terms, so that fact proves that the series is absolutely convergent. May 13 comment How to represent the floor function using mathematical notation? @Nathaniel: We're talking about the $(-\frac\pi2,\frac\pi2)$ branch (the tangent is $\pi$-periodic) :-) Of course it's only convention, but it's the standard mathematical convention to use this branch and not another one. May 13 comment How to represent the floor function using mathematical notation? @Clayton and agksmehx: Using this as an implementation for the floor function would be ludicrous (pardon my French). I'm pretty sure that any implementation of $\tan$ involves the floor function in some way of another.