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Aren't these questions similar?

Factorial Proof by Induction Question: $ \frac1{2!} + \frac2{3!} + \dots+ \frac{n}{(n+1)!} = 1 - \frac1{(n+1)!} $?

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$

I flagged the first one as duplicated but it is declined. Why? I know that it happens when three users voted to let it open. But why they do that is the exactly same.

edit1: I flagged the first one.

edit2: There is also another question that is exactly the same see: Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ I think the other others should be duplicate of this also the first one because this is older than the others and it has more up votes.

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  • $\begingroup$ There should be a reason included on why it was declined. Did you read it? $\endgroup$
    – J. M. ain't a mathematician
    Jul 10, 2016 at 9:13
  • $\begingroup$ @J. M.There isn't anything. $\endgroup$
    – Taha Akbari
    Jul 10, 2016 at 9:14
  • $\begingroup$ @J.M. There's a decline-reason if and only if the flag is declined by a moderator. If the flag is declined via review-queue actions, there's just the outcome, no reason. $\endgroup$
    – Daniel Fischer
    Jul 10, 2016 at 9:19
  • $\begingroup$ In which case, @Daniel, I would submit that it is a rather annoying feature. How is the flagger supposed to know how s/he messed up? Or worse, if the decliners were actually in error because they couldn't be arsed to check? $\endgroup$
    – J. M. ain't a mathematician
    Jul 10, 2016 at 9:22
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    $\begingroup$ I can't tell you why the reviewers voted to leave open, but I agree that the two are duplicates. $\endgroup$
    – Daniel Fischer
    Jul 10, 2016 at 9:22
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    $\begingroup$ I agree with Taha Akbari. The two questions are undoubtedly duplicates. Daniel beat me to closing the later (and less upvoted) version as a duplicate of the other while I was having fun commenting that IMO calculating the value of a telescoping sum is a proof by induction :-) $\endgroup$
    – Jyrki Lahtonen
    Jul 10, 2016 at 9:27
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    $\begingroup$ @J.M. Indeed that's not ideal. But having moderators vet every flag that would be review-declined wouldn't fly on SO (and even here, that would create a lot of additional work for the moderators). But having a message "declined - from review" or similar might be an improvement over the current situation. $\endgroup$
    – Daniel Fischer
    Jul 10, 2016 at 9:33
  • $\begingroup$ None of that sort, @Daniel. :) A "declined - from review" plus a link to the particular review page for that flag was what I had envisioned. But I am loathe to ask in meta.SE; oh well… $\endgroup$
    – J. M. ain't a mathematician
    Jul 10, 2016 at 9:46
  • $\begingroup$ @J.M. I'm going to search a bit, and if I don't find anything, I may post a feature request over there. $\endgroup$
    – Daniel Fischer
    Jul 10, 2016 at 9:51
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    $\begingroup$ @J.M. Brad Larson beat us to it. $\endgroup$
    – Daniel Fischer
    Jul 10, 2016 at 9:54
  • $\begingroup$ Probably it would be better to choose a duplicate target which is not (proof-verification) request. Like this one. $\endgroup$
    – Martin Sleziak
    Jul 10, 2016 at 10:40
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    $\begingroup$ BTW here is link to the review. $\endgroup$
    – Martin Sleziak
    Jul 10, 2016 at 10:47
  • $\begingroup$ A reason could be they did not realize it was a duplicate closure that was proposed, and they thought it was a regular one. This is not unheard of. $\endgroup$
    – quid Mod
    Jul 10, 2016 at 11:43
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    $\begingroup$ There are many possible explanations. (All of them pure speculation, unless they come directly from users who voted.) Here are two possibilities: 1) Proof-verification questions are usually closed as duplicates only after the OP have received sufficient feedback. So it is customary to leave some time. 2) If the reviewers opened the question in new window, they saw that there are no close votes. (The reason is that it was flagged, not close-voted). But they might have been suspicious that it is a review audit, so they decided to stay on the safe side. $\endgroup$
    – Martin Sleziak
    Jul 10, 2016 at 12:12
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    $\begingroup$ "If the reviewers opened the question in new window, they saw that there are no close votes. (The reason is that it was flagged, not close-voted)." - I admittedly did not account for that mechanism, @Martin; so in this case, it behooves the flagger to leave a comment that s/he thinks this is a dupe. A slight annoyance, admittedly, but I suppose we have to make do. $\endgroup$
    – J. M. ain't a mathematician
    Jul 10, 2016 at 12:15

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You have mentioned that you know this means that the flag was declined based on voting in the review queue. It is possible to find even specific review in question. You can find it either manually looking at review queue history. Probably more efficient way is to use this SEDE query. However, since data in data explorer are only updated once a week, usually you have to wait some time before you can find the review in this way.

But I think you do not learn much form the review, only how many users agreed with you or disagreed with you. Probably the only relevant information is whether the post already went through the review queue, but this is only additional information in the case of close votes - since this was a close flag, you have already seen that it was declined. (Looking at review is more useful in the case of suggested edits, where the users who rejected the suggested edit also provide the reason why they chose this option. But for suggested edits it is easier to find the review from your profile; they are listed in the activity tab among suggestions.)

You also ask why the users chose the option to leave open. We can only speculate, but there were some suggestions in comment what the reason might have been:

  • One explanation might be that they simply did not check the proposed closure carefully enough.
  • Another possible factor might be that both questions are tagged . This is a special class for questions, where the OP does not ask for any proof of the given statement but for critique and possible improvements of their own proof. There were many discussions on meta about this type of question, you can check meta posts tagged if you want to learn more. In the case of closures, the consensus seems to be that such questions should only be closed as duplicates after the OP received sufficient feedback on their own attempt.
  • Another possibility that user who were burned too many times by s are extra careful. So they might open the post in new window to see whether it might be an audit. In this case if a user opened the post in new window, they saw that there are no close votes. (The reason is that it was flagged, not close-voted). For this reason they might have been suspicious that it is a review audit, so they have decided to stay on the safe side and vote to leave open.
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