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

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

, each fraction is less than 1. The product rapidly approaches zero. At (2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14...

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 Because the numerator grows factorially ( ) while

The following graph illustrates the "U-shaped" trajectory of the sequence, highlighting the dramatic shift once the numerator surpasses the constant divisor of 14. 4. Conclusion The sequence (2/14)(3/14)(4/14)(5/14)(6/14)(7/14)(8/14)(9/14...

) act as "decay factors," significantly reducing the product's value before the linear growth of eventually dominates the exponential growth of 14k14 to the k-th power 2. Sequence Analysis

increases beyond 14, each new term is greater than 1. Because the numerator grows factorially ( ) while the denominator grows exponentially ( 14k14 to the k-th power

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