I do not know if the problem has been posed or solved before, but I would like to launch a Polymath project about proving that $$ a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}} $$ is a convergent series which it is assigned the two following sequences in OEIS odd sequences and even sequence. A quite detailed description of some attempts can be found on MSE.

I do not know if the problem has been posed or solved before, but I would like to launch a Polymath project about proving that $$ a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}} $$ is a convergent sequence which it is assigned the two following sequences in OEIS odd sequences and even sequence. A quite detailed description of some attempts can be found on MSE.