I dont understand what the representation of a "suite" of numbers will have to do whith the velocity of the approximation of the "irrational" that "serie" of numbers is suppose to approximate(excuse my very bad english).

whenever i made an approximation of a irrational number by a "serie" i have not use any decimal approximation of the numbers of the "serie" (the serial numbers) but letters and function.

of course some series are very slow and some are quicker (but the real thing to consider is the speed of your computing machine when you want to get The approximation)

So i think you should give a concrete exemple in order to conviced me!

another thing i forgot to ask yesterday is "Is the continued fraction of a number is unique?"

that i can check when i will have got time!