if one wanted to find the 'best' converging sequence, you might go about it by brute force. Seems lots of people beat me to it. http://pastebin.com/iujCgUSu
my $lasterror = 3;
for (1..10000){ my $d = $_;
for (3..40000){ my $n = $_;
if (abs(($n/$d) - pi) < $lasterror){
$lasterror = abs(($n/$d) - pi);
print "\n $n / $d: error $lasterror";}}}
# output
# 3 / 1: error 0.141592653589793
# 13 / 4: error 0.108407346410207
...
# 355 / 113: error 2.66764189404967e-007
That's not a converging series. That's just a sequence of ratios that have successively smaller errors. But there's nothing to relate each ration to the next one.
correct - one of observations in the article was that many of the series identified so far lack the magical 355/113 - I was curious what a converging series would look like, not the equation that describes such a series.
I would be interested in identifying this next super series in terms of an equation but given the few minutes on break, and that total it is unlikely someone as bad at math, such as myself, could come up with this series in a few minutes - I opted for the fun programming route.
You still seem confused. What you have is not a converging series, period. You just have a list of successively-closer ratios. A series (in math) is the sum of the terms in a sequence (which itself is a discrete function). You don't have a discrete function, and you definitely don't have the sum of anything.
He does have a converging sequence, though. Many people don't understand the technical (but obvious to mathematicians) distinction between sequence and series.
