The general term of the product can be expressed using factorial notation:
. 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...
The behavior of the sequence is dictated by the ratio of successive terms: The general term of the product can be
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 Introduction The behavior of the sequence is dictated
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.
, each fraction is less than 1. The product rapidly approaches zero. At