One thing which we should keep in mind is that duplicates might be useful. They improve the chance that another user will find the question, since with each duplicate another copy with somewhat different phrasing of the title is added. So if you spent reasonable time by searching and did not find your question, it is not such a bad thing if it is later closed as a duplicate. (And if it is closed as a duplicate, that is not automatically a reason to delete the question.) But at least for basic questions, which have already been asked many times it is probably better to avoid adding new copies. And, of course, trying to avoid duplicates in definitely not the only situation when it is useful to be able to find a questions on this site.
One another thing worth mentioning is that experience both with this site and with the subject of the question helps a lot. A regular user of the site might immediately spot: This is a question I have seen here before. Somebody who has been teaching calculus for a long time can immediately recognize that a question is in fact a standard exercise which is in almost every introductory textbook on the subject and therefore it is very likely to already exist on the site. Experience with the subject can also help to come up with alternative phrases which might appear in the title of question we are looking for. Experience with the site can help in guessing what tag might have been used if this was asked in the past.
But enough side remarks, let us get to the core of the question. However, I should say in advance that you will find here nothing special. The whole answer could be summarized as: use google, use built-in search, have a look at lists of questions (created automatically by SE software or manually by users). I will try to list some useful techniques and I will try also to illustrate them on examples. (And I will expand this post if I am able to think of some other useful tips or if I can think of some other useful examples.) When listing these examples, I will say which posts I was able to find. Of course, as the contents of the site changes, the result of searches might change, too. (And I am also aware that google does not give the same result to everybody, they might slightly differ.)
Let us start with basics:
- You can use external search engine; for instance, you can use Google
and add
site:math.stackexchange.com to restrict the search on this site.
- You can use built-in search on this site. It is good to know that you can refine search using tags. And also some other advanced search tips, which can be found here or here.
Now let me list some examples of things I use when searching for questions.
When searching by tags, it is possible to choose frequent tab. (Which shows questions which have most links. This is good proxy for "questions which are asked frequently".)
Examples:
It might be useful to check list of related questions (and other lists generated by the software)
When viewing a question, you can see list of related questions in the sidebar. (I'd say, based on my experience, that they seem to depend mostly on the title and tags. However I do not know the exact mechanism how they are generated.) For questions, which are asked frequently, you can find out that it is a duplicate surprisingly often just by looking at related questions. And this list can be useful if you search for some question, too. If you managed to find at least a question which has some similar mathematical expression in the title, there is a chance., that you will find something useful among related questions.
There is also another list which is autogenerated. When posting a question, as soon as you write the title, you are shown list named "Questions that may already have your answer". So maybe if you are searching for something, one thing worth trying is to click on "ask question" and just fill the title (and perhaps tags - I am not sure whether they influence this list) and then check the questions which are offered by the software. Some users say that this way of searching returns much better results than using built-in search. See Why are “Questions that may already have your answer” search results better than the actual search results? and Link up the excellent search engine that gives “Questions that may already have an answer” with the search box. (This probably depends on the type of the question and of how large part of the question can be included in the title. And this is another thing which shows that using descriptive titles is a very useful thing to do.)
For some basic questions, which appear here frequently, some users compiled lists containing those questions and links to posts on this site. So it might be useful checking whether your question is there:
Google Images might be useful, too.
Many posts on MSE contain nice images illustrating the problems or solutions. If you have seen before that the problem you are looking for has picture proof and you think that you can recognize the relevant picture, searching on Google Images might be a good way to go.
For example if you have seen that the proof that harmonic numbers grow approximately as logarithm can be illustrated using nice graph, you can try to search for harmonic series site:math.stackexchange.com or harmonic number site:math.stackexchange.com or harmonic logarithm site:math.stackexchange.com (or some other similar possibilities).
If you know that the inequality $\sin x<x$ can be illustrated using a picture you can try
inequality "sin x" site:math.stackexchange.com.
If you recall that you have seen somewhere a proof of Young's inequality illustrated by a nice picture, you can try to search for young inequality site:math.stackexchange.com or young inequality area site:math.stackexchange.com.
How to search for mathematical expressions?
EDIT: Notice that this was written before I learned about Approach0 search engine. There still might be situations where it might be useful to try Google. (For example, when searching for multiple expressions, Approach0 uses OR operator, in Google you can search for posts which contain all of the given keywords. So I can imagine trying Google if I want to find some combination of formula with specific keywords. Or if I want, for some reason, rely on Google's PageRank.) And it is useful to know about several methods of searching.
But generally when searching for formulas on this site, Approach0 is currently my first choice. You can check yourself whether results in Approach0 are good for the examples I listed - see the links at the end of the post.
As soon as we are looking for mathematical formulas, the things become more complicated. (Although I have seen posts, both on meta and on main, mentioning some search engines designated specifically for searching mathematical expression, I have not experimented with them much.)
Posts on this site (mostly) use MathJax (LaTeX) syntax to write mathematics. (And, of course, this is not the only site using MathJax. You might try omit restriction to this site in the searches below to see that you also get results from some other sites.) So we could simply search for parts of the formula we are looking for, and write them using MathJax.
One problem with this is that for many things there are different possibilities how to write the same thing. For example \frac 1x $\frac 1x$, \frac1x $\frac1x$,, dfrac 1x $\dfrac 1x$, or {1\over x} ${1\over x}$.
Another problem is that there are many possible choices for the variables. But still it is not always that bad. Usually if a variable is index in a sum, it will be i, j, k or n. The variables often used in integrals are $x$, $y$ and (mostly after substitution) $t$ or $u$. Terms of sequences will most frequently be denoted as $x_n$, $y_n$, $a_n$, $b_n$. (Maybe sometimes $x_k$ or $x_i$ instead of $x_n$.) In calculation of limits, $n$ is often used for limits of sequences. For limit of functions, the variable is often $x$. So there might be several possibilities, but we might at least try the ones which are most frequently used. (Depending on how much time we are willing to spend with the searching.)
If the formula/object/theorem I am looking for has a name and I know this name, it might be useful to add it among the keywords used in the search. If I can guess some result or technique which might probably be used in the solution, that might give me other reasonable keywords to add.
Even if the search does not return exactly the question we want, but at least something which has similar title, it might still be worth trying to click on that post and checking list of related questions in the sidebar. (Since the list of related questions is highly dependent on the title and the tags, we have a chance that among the questions with similar titles is the question we want.)
I do not know some good general advice to add here. Rather than that, let us try some specific examples.
Some further examples can be found here and here
We want to search for the formula for $\sum\limits_{k=1}^n k^2$
We can try searches like:
sum "k^2" site:math.stackexchange.com,
sum "j^2" site:math.stackexchange.com,
sum "i^2" site:math.stackexchange.com or
sum "n^2" site:math.stackexchange.com.
Already the first search returns: Prove that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
It is probably not too surprising that this was easy to find - I chose a very well-known formula as an example. (I shown above that it is relatively easy to find a post about this formula using tags and frequent tab.) But still a few comments which might be useful in general, not only for this particular question.
If I know what the result is supposed to be, that might help me further limit the search results. For example, I could search for something like: sum "k^2" frac "2n+1 6" site:math.stackexchange.com.
In this case, searching for the textual description would also help. Search for
sum first squares site:math.stackexchange.com returns How to get to the formula for the sum of squares of first n numbers? and also some other relevant result. (Notice that the post I linked to is closed as a duplicate. But the existence of the duplicate still helps when searching.)
We want to find posts about $\sum\limits_{k=1}^\infty\frac 1{k(k+1)(k+2)}$.
A reasonable thing to try is: sum frac "k(k+1)(k+2)" site:math.stackexchange.com. We might also try other names of variables and search for sum frac "n(n+1)(n+2)" site:math.stackexchange.com or
sum frac "i(i+1)(i+2)" site:math.stackexchange.com or sum frac "j(j+1)(j+2)" site:math.stackexchange.com.
If we can guess what could be used in the proof, we can have some additional phrases which might improve our chances to find posts about this. (If we guessed correctly and the added word is indeed used in on of the answers.)
For example, telescoping sum frac "k(k+1)(k+2)" site:math.stackexchange.com or
partial fraction sum frac "k(k+1)(k+2)" site:math.stackexchange.com or induction sum frac "k(k+1)(k+2)" site:math.stackexchange.com
Almost any of the searches I listed about return some relevant results - either about the infinite series or about the finite sum $\sum\limits_{k=1}^n\frac 1{k(k+1)(k+2)}$. The latter is also helpful for somebody interested in this question.
We want to find posts containing the limit $\lim_{n\to\infty} (\sqrt{n+1}-\sqrt n)$.
We can search for limit "sqrt n+1 sqrt n" site:math.stackexchange.com
Among the first results I found this post Convergence of $\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$ - it is about different question, but contains the computation of the limit. (So this was not entirely unsuccesfull attempt.)
We could also try limit "sqrt n sqrt n-1" site:math.stackexchange.com, since the limit might be shifted by one.
Finally after tryging
limit "sqrt x+1 sqrt x" site:math.stackexchange.com
I found this question Limit problem: $\sqrt{x+1} - \sqrt{x}$ as $x$ approaches infinity (And some other posts about the same limit are linked to it.)
EDIT: Here you can try to search for the same expressions in Approach0:
Since the above examples include infinite series and limits, I will point out that Approach0 returns different results if you replace $\infty$ by $+\infty$. (And both notations are relatively common, especially for limits.) You can test this if you search for $\sum\limits_{k=1}^{+\infty}\frac 1{k(k+1)(k+2)}$ and for $\lim_{n\to+\infty} (\sqrt{n+1}-\sqrt n)$.
