Activity for r~~
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #282665 | Initial revision | — | over 3 years ago |
| Answer | — |
A: $\sum_{k=0}^{n} \binom{n}{k}=2^{n} \overset{?}{\iff} \sum_{k=0}^{n} \binom{2n+1}{k}=2^{2n}$ The green result can be used to prove the red one, yes. Start with the fact that $\binom{m}{k} = \binom{m}{m - k}$. If this isn't obvious to you, you can derive it either by looking at the usual formula for the binomial, or by considering that choosing the members of a subset is equivalent to choo... (more) |
— | over 3 years ago |
| Edit | Post #282654 | Initial revision | — | over 3 years ago |
| Answer | — |
A: How does the change of variable $\color{red}{r↦n−r}$ transmogrify $\sum\limits_{r=0}^n2^{n-r}\binom{n+r}{n}=2^{2n}$ into $\sum\limits_{k=0}^{n} \frac{\binom{2n-k}{n}}{2^{2n-k}}=1$? $n$ is free in the expression $\sum{r=0}^n 2^{n-r}\binom{n+r}{n}$. The only variable not free in that expression is $r$. What $\sum{r=0}^n$ means is to treat what comes next as a function of $r$, and compute the sum of that function evaluated at $0$, $1$, and so on up to $n$. If $n$ is used inside th... (more) |
— | over 3 years ago |
| Comment | Post #282652 |
The variance argument is only relevant if you draw from the same urn multiple times, which isn't part of the original problem. With one draw, the author is correct that you should be indifferent. Your argument is interesting additional information, but I think it's misleading to present it as a count... (more) |
— | over 3 years ago |
| Edit | Post #282638 |
Post edited: |
— | over 3 years ago |
| Edit | Post #282638 |
Post edited: |
— | over 3 years ago |
| Edit | Post #282638 |
Post edited: |
— | over 3 years ago |
| Edit | Post #282638 | Initial revision | — | over 3 years ago |
| Answer | — |
A: A formal-logic formula for decimal to binary conversion If you have a non-negative integer $n$, and the base $b$ representation of $n$ is a sequence of digits, numbered from the right as $dk\dots d2d1d0$, then $di = \left\lfloor \frac{n}{b^i} \right\rfloor \bmod b$. For example, decimal 10 is binary 1010 because: $$ d3 = \left\lfloor \frac{10}{2^3}... (more) |
— | over 3 years ago |
| Comment | Post #282615 |
$\ldots$ doesn't work like that; you can't simply substitute a variable into that kind of informal, descriptive expression without thinking about what the notation as a whole represents. It's not an algebraic expression, so the rules of algebra don't apply. $\ldots$ is a signal to look at the pattern... (more) |
— | over 3 years ago |
| Comment | Post #282623 |
I think the feedback given in the other thread is better. Citing edition and page number doesn't make the question any clearer; it just makes it possible for someone to do detective work to fill in details that the asker should have provided in the first place. An image of the page doesn't lead to a ... (more) |
— | over 3 years ago |
| Comment | Post #282622 |
Related: https://math.codidact.com/posts/280856#answer-280857 (more) |
— | over 3 years ago |
| Edit | Post #282625 | Initial revision | — | over 3 years ago |
| Answer | — |
A: Questions with a quote/screenshot and a request to explain My approach is downvote and, if it's a first infraction, comment explaining the issue with the question. Repeat offenders get downvotes and, perhaps if this one-guy's-opinion calcifies into actual community policy, flags. I'd like us to start doing a better job of holding the line against low-effo... (more) |
— | over 3 years ago |
| Comment | Post #282616 |
An understandable mistake, but one you perhaps could have caught yourself if you had thought about your own interpretation a little more. $k$ doesn't make sense as the number of ways to choose $k$ team members, any more than $\(^n_k\)$ makes sense as the number of ways to choose one of them to be cap... (more) |
— | over 3 years ago |
| Comment | Post #282611 |
We've had words before about the appropriateness of screenshots in questions. In this case, the quoted text doesn't seem to be relevant to your question. The question is about the screenshotted text, which is completely different, and from an online source which I seem to be able to copy from just fi... (more) |
— | over 3 years ago |
| Comment | Post #282615 |
I don't even know what the middle expression in that line is supposed to mean. You start at $n$, and then descend to $n - 1$, and by the end somehow... wrap around to $n + 2$, $n + 1$? What's happening in between, where the ellipsis is? How did you arrive at that as what $LHS\|_{k=1}$ should represen... (more) |
— | over 3 years ago |
| Edit | Post #282616 | Initial revision | — | over 3 years ago |
| Answer | — |
A: Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"? $\(^nk\)$ is the number of ways to choose $k$ team members from $n$ players. $k$ is the number of ways to choose one captain from $k$ team members. (It isn't the number of ways to choose a captain from the original group of people, as you suggested; that would be $n$.) $k\(^nk\)$ is the product... (more) |
— | over 3 years ago |
| Edit | Post #282615 | Initial revision | — | over 3 years ago |
| Answer | — |
A: If k = 1, why $n(n-1) \dots \color{red}{(n-k+1)} = n$? You started on the right track with $n(n - 1)\ldots(n - [k - 3])(n - [k - 2])(n - [k - 1])$. What you should have noticed is that there are $k$ terms in that product. The first term subtracts 0 from $n$, the second term subtracts 1, and so on, with the $i$th term subtracting $i - 1$, all the way u... (more) |
— | over 3 years ago |
| Comment | Post #282611 |
You screenshotted a website? Is this an elaborate troll or something? (more) |
— | over 3 years ago |
| Comment | Post #282609 |
In a spree of low-quality questions, this one stands out as exceptional. (more) |
— | over 3 years ago |
| Suggested Edit | Post #282600 |
Suggested edit: (more) |
helpful | over 3 years ago |
| Edit | Post #282027 | Initial revision | — | over 3 years ago |
| Answer | — |
A: Why mustn't the proportion of smokers among married people be the same as the proportion of smokers in the whole population? The key word is exactly. If I flip a fair coin, I expect about half of my flips to be heads. If I flip it twice, exactly one head is quite likely. If I flip it twenty times, exactly ten heads is not that unusual, but still a little lucky maybe. If I flip that coin one million times and get exactly 50... (more) |
— | over 3 years ago |
| Comment | Post #282015 |
Completely off topic, but: if you've been taught that, in English, choosing a more obscure word always demonstrates mastery of the language, you've been taught incorrectly. Inappropriate use of obscure words looks far sillier than common words used correctly. (If you are fortunate enough even to find... (more) |
— | over 3 years ago |
| Comment | Post #281724 |
Does your kid know about 30-60-90 triangles? Knowing that the short side is half the length of the hypotenuse would be enough. (more) |
— | over 3 years ago |
| Edit | Post #281638 | Initial revision | — | over 3 years ago |
| Answer | — |
A: difference between quotient rule and product rule If you work out deriving the quotient rule yourself using the exact trick you're highlighting, you can see that the quotient rule is nothing more than the product rule and the chain rule used together. If you rearrange a quotient to use the product rule, then you'll very likely be using the chain rul... (more) |
— | over 3 years ago |
| Comment | Post #280716 |
Never claimed otherwise! In my first paragraph, I specifically referenced ‘multiplication of square matrices’ as an example of ring multiplication. Cross products are an example of my third case, things that don't conform rigorously to ring multiplication or category products, but which borrow the na... (more) |
— | over 3 years ago |
| Comment | Post #280842 |
_Counterexamples in Analysis_ (Gelbaum and Olmsted), p. 126, offers $\vec{F}(x, y, z) = (x^2 + y^2 + z^2)^{-\frac{3}{2}}(x\vec{i} + y\vec{j} + z\vec{k})$, defined everywhere but the origin, as a counterexample. This suggests that a simply connected domain is not sufficient—IIRC it needs to be 2-conne... (more) |
— | almost 4 years ago |
| Comment | Post #280857 |
I would say, in addition to ‘is also typed into the post’, that the text should be understandable without the image—so, in particular, having the main question of the post refer to a circled part of a screenshot can still be a problem, even if that circled part appears elsewhere in the post (because ... (more) |
— | almost 4 years ago |
| Edit | Post #280854 | Initial revision | — | almost 4 years ago |
| Answer | — |
A: How can I visualize this Compound Interest Chart with indefinite integrals? Neither of those charts presents those integrals; the first illustrates annually compounded interest, and the second compares a few different compounding schedules to continuous compounding, but not in a way that highlights the integrals in the text explanation. If you wanted an illustration of th... (more) |
— | almost 4 years ago |
| Comment | Post #280847 |
In financial math, the term ‘point’ is used precisely to indicate what you call ‘absolute increase’. The difference is that everyone in the field will know what you mean if you say ’basis point’ or ‘percentage point’, and people will ask for clarification if you say ‘absolute increase’. Your wording ... (more) |
— | almost 4 years ago |
| Comment | Post #280851 |
That won't do; that definition requires the limit to be a finite number, but you're working with an infinite limit.
Also, downvoted because once again, you're using images of text instead of writing your question such that no images are required. (more) |
— | almost 4 years ago |
| Edit | Post #280799 |
Post edited: |
— | almost 4 years ago |
| Suggested Edit | Post #280799 |
Suggested edit: (more) |
helpful | almost 4 years ago |
| Comment | Post #280717 |
The graph in the upper-right corner of the proof box is all you need. Imagine that's a graph of $f_1$. Once you have $f_1$, the details of $f$ and $g$ don't matter and graphing them is neither necessary nor useful, though you can always imagine what that would look like if you want. (more) |
— | almost 4 years ago |
| Edit | Post #280717 | Initial revision | — | almost 4 years ago |
| Answer | — |
A: How can I generalize a picture for the Mean Value Theorem to the Generalized MVT? Plotting $f$ and $g$ together is unlikely to lead to intuition about the GMVT; they're just two arbitrary curves, whereas the value from that diagram of the MVT comes from the fact that the yellow line is constructed to give intuition. The art of drawing intuitive figures is largely about figuring ou... (more) |
— | almost 4 years ago |
| Edit | Post #280716 | Initial revision | — | almost 4 years ago |
| Answer | — |
A: Why should a non-commutative operation even be called "multiplication"? Usually (not always, but true for the examples of multiplication of real numbers and multiplication of square matrices), a binary operation that is called some variant on ‘multiplication’ is the multiplicative operation of some ring). That's the closest thing to a rigorous definition of multiplicatio... (more) |
— | almost 4 years ago |
| Edit | Post #280168 |
Post edited: Fix TeX formatting |
— | almost 4 years ago |
| Comment | Post #280206 |
I suspect $k$ is an idiosyncratic or just unfortunate choice of independent variable. It looks like the author is just saying that $\mathcal{M}$ is the identity function, like $f(x) = x$—at least, I don't see any reason to say anything more nuanced than that about $\mathcal{M}$, given what's discusse... (more) |
— | almost 4 years ago |
| Edit | Post #280204 |
Post edited: Fix TeX formatting |
— | almost 4 years ago |
| Edit | Post #280206 | Initial revision | — | almost 4 years ago |
| Answer | — |
A: Given a triangle with squares on two sides, the line segments joining the centres of the squares to the midpoint of the third side are equal and perpendicular Your calculations look correct to me, but I think this is perhaps meant to be seen geometrically (apologies if this is already obvious to you and you were looking for a non-geometric explanation). Let the vertices of the triangle be $x$, $y$, $z$, starting at the vertex shared by the two squares, ... (more) |
— | almost 4 years ago |