"The Principle of Shrinkage"

[Henry]

How about lack of convergence of the Stirling series (A046968 
<http://www.research.att.com/%7Enjas/sequences/A046968> and A046969 
<http://www.research.att.com/%7Enjas/sequences/A046969>). I cannot be 
certain of the date the series was put forward but Stirling's /Methodus 
Differentialis /was published in 1730 and Bayes's letter in 1763.

Edwin Clark wrote:

>  If f:R -> R or f:N -> R is a "naturally occurring function" and
>  lim(x->oo) f(x) appears to exist, then it does.
>
>Then Tim asks the question:
>
>Can anyone give an example of a conjecture of the form "lim(x->oo) f(x)
>exists" for which there was extensive numerical evidence and which
>remained open for some time, but was eventually shown to be false?
>
>  
>