Convergence of an=(1−12)(12−13)…(1n−1n+1)a_n=(1-\frac12)^{(\frac12-\frac13)^{…^{(\frac{1}{n}-\frac{1}{n+1})}}} [closed]

I’m interesting to see the opinion of MO about my question which I posted here in SE, Answers I received have not convinced me, And no clear proof posted there only numerical computation are provided. I should post the question as it is in order to know more about convergence of the titled sequence

let us moving to telescopic sum using exponent, Assume we have this sequence: an=(112)(1213)...(1n1n+1) with n1, this sequence can be written as power of sequences: xnyncn such that all them value are in (0,1), I want to know if the titled sequence should converge to 1? and how we can evaluate it for n go to ?


Source : Link , Question Author : Zeraouli Rafik , Answer Author : Community

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