The guidance for the convergence-divergence and sequences-and-series tags are as follows:
convergence-divergence
Tag guidance:
Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.
Tag wiki:
This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.
Formally, a sequence $S_n$ converges to the limit $S.$
$$\lim_{n\to \infty}S_n=S $$
if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.
A sequence diverges if it does not converge.
For more specialized questions about divergent series, such as summation methods, consider the tag divergent-series.
sequences-and-series
Tag guidance:
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
Tag wiki:
Sequences and series are often considered as a main part of part of calculus, in addition to limits, continuity, differentiation and integration.
A sequence is an enumerated collection of objects in which repetitions are allowed. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous; harmonic progressions, which is formed by taking the reciprocal of each term of an arithmetic progression; logarithmic progression, which is formed from a series whose progression getting smaller; and geometric progressions, where the next term is a constant multiplied by the previous term.
A sequence can be given by a direct formula (e.g. $a_n = 2^n + 3$), or by a recurrence relation. In a recurrence relation, the relation between the next term and the earlier terms is given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n, n \geq 0.$ Together with the initial terms $F_0=0$ and $F_1=1$, this recurrence relation defines the famous Fibonacci sequence.
A series is formed by summing a sequence. A typical question is: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is it convergent? Several tests, such as the ratio test, the root test, the limit comparison test, the integral test, etc. can help to answer these questions.
Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting these questions at Puzzling Stack Exchange instead.
While this isn't explicitly stated in the tag descriptions, I think that it is implicit that these tags are primarily meant for the kinds of questions which might arise in an elementary calculus class:
The sequences-and-series tag is almost explicit about this, and notes that the topic of the tag is a core part of calculus (along with differentiation and integration).
The convergence-divergence tag is less explicit, but references convergence tests and improper integrals, which suggests an elementary calculus class.
As such, I would argue that neither tag is really quite appropriate for a question about Cauchy sequences in a general metric space. Anyone who is asking questions about Cauchy sequences is likely taking a course in real analysis (or, perhaps, point-set topology? a lot of elementary topology texts seem to start from an analytic / metric point of view, e.g. Munkres), and is not asking a basic question about sequences and series, but is asking a more advanced question.
Hence I think that the removal of these two tags is not unreasonable. The general-topology tag might be relevant (depending on the question itself), and the metric-spaces tag might also be helpful. The cauchy-sequences tag would almost certainly be relevant.
Question for Discussion:
Should the sequences-and-series and convergence-divergence tags be reserved for more elementary calculus questions?
If yes, this should probably be made more explicit in the tag wikis.
