Instead of reasking the remaining question in case of a partial answer, one can also edit the original question and state which parts have been answered by the answers already given, and which parts are still open.

@TobiasKildetoft: I would write that very explicitly. -- People tend to interpret the question in their favorite way otherwise, as it happened e.g. with my question Mathematical research published in the form of poems, where answerers freely assert that e.g. medieval or ancient chinese poems have been published in a similar form as a paper in Math. Z..

I would welcome such a question -- though I think you should give examples of the mentioned failed attempts at proving / disproving the amenability of $F$, in order to illustrate what you are asking.

I cannot reproduce this problem.

@ToddTrimble: My impression is that only a pretty low percentage of the questions which get closed on MO are closed in error (it takes five votes to close a question, and anybody may post a comment pointing out that the question is suitable for MO, and be it only "why the close votes?") -- and if a question is closed, it may still be reopened. I think if people would cast close votes only when they are absolutely sure that there is nothing good in a question, it would probably be notably more difficult to keep the site on-topic.

Well -- I think this depends on the particular case. Posting a new question just because one has forgotten to exclude some trivial case seems not reasonable to me -- in that case I think it is best to fix the question and the commenter who has highlighted the omission removes their now outdated comment.

I think this would be something to decide for the moderators or the SE staff.

Presumably the same person has posted also a lot of other junk, so it doesn't seem this is targeted at you. I think this would be a case for blocking the offender's IP number(s). By the way, downvoting requires 125 points -- see mathoverflow.net/help/privileges.

Taking an arbitrary convergent series, typically its value has no "closed form expression". -- Do you know any reasons why for your series such expressions should exist?

Why are you interested in this particular series, and what kind of answer would you expect -- a numeric approximation, or what else?

"... and ask if someone has seen it or has something to say about it": do you have in mind a reference request, or are you rather interested in people's opinions, or in hints on further directions of research people may give you, or where else is the focus of your question?

@GottfriedHelms: That's a good point. -- Hosting valuable content on an ad server is probably not a good choice.