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

), Stirling's Approximation confirms that the product will ultimately diverge to infinity. 3. Visualization of Growth

. 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 (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 ), Stirling's Approximation confirms that the product will

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 We analyze the transition point where the sequence

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

is a classic example of a sequence that appears to vanish but eventually explodes. While the initial terms suggest a limit of zero, the "power" of the factorial ensures that for sufficiently large , the product overcomes any constant denominator.

The behavior of the sequence is dictated by the ratio of successive terms: