# List of Generalisations of Common Questions

Here is a list of generalised s to which other questions may be deduped against, split by topic (please edit the question).

# Graph theorygraph-theory

How to tell whether two graphs are isomorphic?

# Group theorygroup-theory

For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?

An element of a group has the same order as its inverse

How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?

# Polynomialspolynomials

• Shouldn't the questions be "answers" rather than edits to the question? Mar 31, 2011 at 18:27
• @Arturo: Yes, we could have one answer per category. Having one question per answer might not be that good. The intent was to keep the list in one place so that we can browse through easily. Mar 31, 2011 at 18:30
• @Moron: Yes, I understand; but (i) we probably don't want this question to keep popping up as "unanswered". And (ii) I think if we had one answer per category (or per tag) and have people "add it to the list" there it would make more sense. Right now the question is fine, but if this keeps up, the question itself will grow too lengthy to be manageable, I think. Mar 31, 2011 at 18:32
• @Arturo: I think having this question pop up periodically is probably a good thing :-) I do agree with one tag answer thing. If enough people upvote your comment, I will make the edits :-) Mar 31, 2011 at 18:37
• @Arturo: I think that questions from the unanswered list only get bumped if they have an answer (necessarily none accepted or with positive vote count). (If I'm wrong, will someone please inform me?) But I agree with (ii). Mar 31, 2011 at 18:44
• @Isaac: But that might lengthen the list by a large amount. I mean both the questions in the calculus tag have other tags, namely "Integral" and "Limit," but I don't think it is a good idea to create separate categories and put them in those categories. That wouldn't be a useful addition. The geometric series is definitely in the series category, why does it need to be in algebra/pre-calculus as well? Tagging the question as such is of course a good idea. Then searching the FAQ tag on the main site is very easy. But in this list? I don't understand the need. Mar 31, 2011 at 20:04
• Question: Should questions such as "Why is $0.999..=1?$" "Why is $0^0=1$? and "Why does $0!=1$" be added to this list? Should we make one question dealing with all of them at once? I mean they are asked frequently enough, occasionally with slight variations, but are definitely not "abstract generalizations." Thoughts? How could one question be added that deals with all of these type of things at once? Mar 31, 2011 at 20:35
• @Jeff Atwood: I think it'd be better for each tagged subject to have a corresponding meta thread of faqs, abstract duplicates, etc. These could be displayed to low-rep users when they add a tag to a question. Better we could force them to choose a subject/tag before composing a question, so they see the subject-specific faqs first. This would go a long way towards removing the noise/overhead generated by duplicate questions (a big problem for a general-level math forum since exercises are often tweaked year-after-year to eliminate copying - yielding abstract, not exact, duplicates). Apr 8, 2011 at 20:19
• @Arturo: I am waiting for 2 more upvotes to your comment :-) Apr 8, 2011 at 20:49
• I don't understand why geometric series are under two categories. Apr 17, 2011 at 19:25
• math.stackexchange.com/questions/8337/… should be added, as questions about $\zeta(2)$ arise frequently.
– Asaf Karagila Mod
Apr 20, 2011 at 16:02
• Maybe we should add this to the list (at least until this silliness dies down and people move on to other things)? Apr 29, 2011 at 17:52
• May be this and this should be brought under the common umbrella of obvious consequences of the binomial formula? Sep 27, 2011 at 10:23
• Hmm, I actually think it's a good thing that this "question" has no upvoted answers (and to that effect, please don't upvote any!). The bot will bump it up periodically, and we always get a reminder of these dupes. Dec 19, 2011 at 0:28
• @PeterT.off: Please don't forget to edit the question 1) to include the full general problem. 2) state that it is being repurposed (see other such questions from list above for a template which you can cut and paste) 3) and tag it as (faq). Apr 20, 2012 at 1:22

• Let's try to keep these in order by post ID number so that it is easy to avoid duplicates.
– MJD
Apr 8, 2014 at 2:39
• How would I create a tag "Polya/Burnside" on MSE and if possible re-tag these questions automatically? Apr 28, 2014 at 23:26
• @MarkoRiedel Tags are created simply by adding the new tag to a question. When creating tag, it is good to write also a tag-wiki/tag-excerpt for the new tag. It's up to you whether you discuss creating the new tag on meta first, or whether you create tag without such a discussion. But I think that it is a good thing that those questions will be bumped when retagged - other users will be alerted about the new tag by this and if they disagree, they can react on meta. May 6, 2014 at 8:07
• This is one topic (Polya-Burnside) rather than several. The questions on coloring cubes and counting necklaces get asked all the time over and over again. That's why MJD suggested we make this list. Jun 28, 2014 at 13:53

More for my own sake than anything else. Integration duplicates tend to be quite hard to find due to the large amount of symbols in titles. Here is a short list of some famous integrals that pop up periodically

### Integrationintegration

• Perhaps we could add Evaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$ to the list? Aug 28, 2017 at 21:47
• You can search by MathJaX, the problem becomes how many ways are there to create the same thing.
– user645636
Aug 14, 2019 at 16:55

There ought to be an entry for the / classic:

$$\lim_{n\to\infty} \left(1+\frac xn\right)^n = \exp x$$

I found these posts on it so far:

which I've ordered from most to least suited for status (after a suitable amount of editing). There are probably more, but I couldn't find them by searching the site.

(I've posted this answer because it is not obvious which one of the above should get status.)

• You can add it yourself. Please feel free to do so. Pick one as the representative, and tag that as FAQ. I also downvoted this answer so that this question is periodically bumped up (no answer with > 0 votes). Jan 7, 2014 at 23:07

# Hypergeometric/Summation Identities summation

Rather than have these broken up among algebra-precalculus, proof by induction, combinatorics, etc. I am gathering them all in one place.

### Faulhaber identities (sums of powers)

$$\sum\limits_{k = 1}^n k = \frac{k(k+1)}{2}$$ (triangle numbers)

$$\sum\limits_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ (pyramid numbers)

$$\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$$ (sum of cubes)

$$\sum\limits_{k=1}^n k^p$$ (general formula)

### Geometric series/(Generalized) Binomial Theorem

$$\sum\limits_{k=0}^n \binom{n}{k} (-1)^k$$

$$\sum\limits_{n = 0}^\infty x^n = \frac{1}{1-x}$$ and $$\sum\limits_{n = 0}^N x^n = \frac{1 - x^{N + 1}}{1-x}$$ (geometric series)

$$\sum\limits_{n = 0}^\infty (n+1)x^n = \frac{1}{(1-x)^2}$$ (derivative of geometric series)

$$\sum\limits_{n=0}^{\infty} \binom{n+k}{k} x^n = \frac1{(1-x)^{k + 1}}$$ (Negative Binomial Theorem)

$$\sum\limits_{k = 1}^n k \binom{n}{k} = n2^{n-1}$$ (derivative of binomial theorem)

## Vandermonde-like identities

$$\sum\limits_{k = 0}^{n} \binom{a}{k} \binom{b}{n - k} = \binom{a + b}{n}$$ (Vandermonde's identity)

$$\sum\limits_{k = 0}^{n} (-1)^k\binom{a}{k} \binom{b}{n - k} = \left[x^n\right](1-x)^a(1+x)^b$$ (alternating Vandermonde sum)

$$\sum\limits_{k=0}^{n}\binom{n}{k}^2 = \binom{2n}{n}$$ (special case of Vandermonde)

### Other

$$\sum\limits_{k=0}^{N}\binom{k}{n}=\binom{N+1}{n+1}$$ (Hockey-Stick Identity)

$$\sum\limits_{k=0}^{n}\binom{2k}{k}\binom{2n-2k}{n-k}=4^n$$ (convolution of central binomial coefficients)

# Statisticsstatistics

• Upper tail inequality for the standard normal distribution:

• Normal approximation (CLT) for discrete Binomial: ... (to be continued... okay, this task is more complicated than expected)

• About the topic Expectation as tail probability (complementary CDF) integral/sum. (1) the continuous case, other candidates I've considered are two older posts: 63756, which answer generalizes to "any" $E[g(X)]$, and 64186 that is highly upvoted with several good answers. I went with 172841 because of existing links. It seems that for a while some users agreed to link many duplicates to this one. Nov 13, 2018 at 13:18
• (2) the discrete case, there’s another strong candidate 74186 (better than a runner up that is more recent) that also has two posts closed to it as duplicates (not just linked). I went with 843845 because there are more content and more "natural links". There’s also 660185 that has one post closed to it as a duplicate. Nov 13, 2018 at 13:18
• (3) I’m not sure about having a separate sub-category of measure-theoretic treatment. As I went through the process of dropping comments for all the duplicates I found, I barely linked them to what I chose, the post 402640. In other words, I'm not sure it really deserves the tag faq. Nov 13, 2018 at 13:54
• (4) there are posts with extension (sort of easy) to non-negative r.v. (with absolute value or split integral). I’m not sure if they should have a separate "mother post" here and none of them seem to standout as much better than the others for such purpose. Currently I’ve identified five: 1349624, 1350577, 1427801, 1489123, and 2300167. Nov 13, 2018 at 13:55
• The probabilities for hitting either of two boundaries in a simple symmetric one-dimensional random walk are treated in many posts, but I haven't been able to identify a canonical one with a general treatment that would be a good original for all duplicates. Mar 15 at 6:09
• I just closed a question about this by referring to these five questions, none of which seems optimal: 3497645, 145705, 4253962, 3501693, 4583814. If you find a better fit, please add it to the answer above. Mar 15 at 6:09
• I am sure I have missed some generic frequently referenced methods, or there might better candidate, anyone please update accordingly...
– Sil
Aug 11, 2018 at 11:18

There are some standard questions in general-topology:

### Book recommendations:

A couple of analysis exercises that appear frequently: