(2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14... > (2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14...

(2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14... [VERIFIED]

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Pk=k!14k−1cap P sub k equals the fraction with numerator k exclamation mark and denominator 14 raised to the k minus 1 power end-fraction 2.1 The Critical Threshold (2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14...

. We analyze the transition point where the sequence shifts from monotonic decay to rapid divergence and discuss the number-theoretic implications of the denominator's primality relative to the numerator's growth. 1. Introduction AI responses may include mistakes

R=Pk+1Pk=k+114cap R equals the fraction with numerator cap P sub k plus 1 end-sub and denominator cap P sub k end-fraction equals the fraction with numerator k plus 1 and denominator 14 end-fraction For all (2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14...

, each fraction is less than 1. The product rapidly approaches zero. At