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

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.

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...

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 is a classic example of a sequence that

Infinite products are a cornerstone of analysis, often used to define functions like the Gamma function or the Riemann Zeta function. The sequence presents a unique case where the first twelve terms (for Because the numerator grows factorially ( ) while

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