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7:53 AM

Some people here on MO keep complaining about MO questions appearing in HNQ: Measures to separate math overflow from the rest of the stack exchange network and Featured MO questions on the hot list: what benefits, if any, do these bring?.

But if there is a question with a totally non-descriptive title, nobody edits the title to include the formula in question. (Which would also remove the question from HNQ.)

Q: does this sum have a limit?

Define the sequence given by the finite sum $$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$ Questions. (1) Is $0<a_n<1$? (2) Does the limit $\lim_{n\rightarrow\infty}a_n$ exist? If yes, what is its value?

Related from meta.SE (I learned about this post from quid):

Q: Encourage active users to edit Hot Questions, especially titles

Questions on the "Hot Network" list are like travel billboards: they advertise sites to users elsewhere, to maybe get those users interested in participating. But I have noticed that those questions don't seem to get edited any more than others; bad titles pop up in the list, and they stay. I stu...