No proof yet, but here's what I found.

Let n be the number of terms of the product, and let d be the denominator the fraction x. So in the actual problem, n is 2017 and d is 2018. Here are the outcomes for some smaller values of n and d. Let's look for a pattern. Since 2018 is 2 modulo 4, and 2017 is one smaller than 2018, I will continue to let d be 2 modulo 4 and n be d-1.

n=1, d=2 gives -1

n=5, d=6 gives -1/16

n=9, d=10 gives -1/256

The pattern looks like the denominator is exponential with base 16. So potentially the product for a given d and n=d-1 is -16^-floor(n/4).

With n=2018, floor(n/4) is 504, so my guess at the answer is -16^-504, which is a very small number.

Posted January 20th
by EN