When using (summation) tag, should we remove (sequences-and-series)?

Question

The tag summation was created less than a year ago, see "summation" tag for finite and formal summations

Before that, post about finite sums were usually tagged as sequences-and-series. Since then the new question have been usually correctly tagged as summation, and some of older questions have been retagged too.

The tag-excerpt and tag-wiki for summation say:

Use sequences-and-series for sums of infinite series and questions of convergence; use summation for questions about finite sums and simplification of expressions involving sums.

Based on this it seems that typical question on finite sums should be tagged only summation and not sequences-and-series. (Of course, there might be exceptions where both tags are appropriate.) Yet we have many question tagged with both these tags.

I have to say that I am usually careful with removing the tags the the OP has chosen, especially if I am not entirely sure. (So some of occurrences of both tags are due to my retags.)

I would like to ask on the opinion of the community about this combination of tags. Perhaps this will encourage more people (including me) to use the tags correctly. And if most users disagree with the guide given in the tag-wiki, we can change the tag-wiki.

• Should the tag sequences-and-series be removed the questions about finite sums and only the tag summation should be used from this pair of tags?

Answer 1

When I created summation, it was with the idea that questions about series are usually concerned with issues of convergence, and that summation was a distinct tag for something quite different. A question such as

What is a closed form for $$\sum_{i=0}^n i^2?$$

is about a certain formal, algebraic identity. It has a completely different character from:

What is a closed form for $$\sum_{i=0}^\infty i^{-2}? $$

which is about convergence properties of the real numbers. The answer to the first question is an expression, expressing a particular function of $n$. The answer to the second question is a number, or a proof that the indicated series diverges. Someone who said the second summation was divergent would be wrong. But someone who said the first summation was divergent would be “not even wrong”, but deeply confused about what the question was.

Only the first question should be tagged summation; only the second one should be tagged sequences-and-series. Summations, such as the first example, have nothing at all to do with sequences, and it makes no sense to bundle sequences into their tag. Series, such as the second example, are sequences, and it makes sense to tag them together.

I tried to convey this dichotomy when I wrote the tag descriptions for summation. Unfortunately, the tag descriptions and tag wiki are very hard to find and to navigate.

In my opinion, it should be very rare that a question should receive both tags. I cannot recall any example for which both tags would be appropriate.

Answer 2

In fact the tag wikis of summation and sequences-and-series are not properly scoped, and even they are not named properly.

Sequence is an ordered list of objects (the objects can be numbers, functions, etc.), Series initially it refers the sum of the terms of a sequence, but in modern mathematics it even extended to refer the summation-type kernel function. The complicated type of series can contain several summation signs (e.g. http://en.wikipedia.org/wiki/Kamp%C3%A9_de_F%C3%A9riet_function). Summation only refer the operation of sum, the summation tag has an inappropriate tag name.

The questions about sequences are not necessarily interested about sum of the terms, while the questions about series are often interested about e.g. the existance of close-form, the convergence etc. rather than about e.g. the origin of which sequences, so the sequences-and-series tag also has an inappropriate tag name.

Even the tag wikis of the sequences-and-series tag and the summation tag are very awful.

The tag wiki of the sequences-and-series tag:

Recurrence relations, convergence tests, identifying sequences

Does't there already have the recurrence-relations tag specify for questions about recurrence relations?

The tag wiki of the summation tag:

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

The first sentence and the second sentence are contradicted, since the first sentence said that questions about infinite series can use the summation tag, but the sentence sentence said that can't. Therefore no wonder why quite many people use both the summation tag and the sequences-and-series tag.

So the summation tag and the sequences-and-series tag should be deleted. The replacement should be to create the sequences tag specifically for the questions about sequences, the finite-series tag specifically for the questions about finite series, and the infinite-series tag specifically for the questions about infinite series.