sorry it took a bit, had to type it out in LaTeX to properly write the notation, but i've constructed a proof that there is no '
slowest convergent rate' :
View attachment 223474
so i can always pick an arbitrary n and make any sequence converge arbitrarily slower.
the implied definition of '
most irrational' being used in the bogus things
@cv5 scoured the internet for, trying to save face and make his weird insult stick, reduces to '
slowest convergence rate' -- this is essentially what the difference between the actual number & sums of truncated expansions is measuring.
a slow convergence rate is driven by the ration between consecutive terms being close to 1. i've shown that i can take any convergent series and transform it into an equivalent one with a slower rate of convergence, therefore using the silly definition of 'most irrational' i can create an infinite series for any number, rational or not, which converges at an even slower rate than any given convergence rate.
i.e. using this definition i can make any rational number more irrational than any irrational number.
QED it's a poor, unworkable definition.
not that it matters because even without this proof there are infinite multiples of φ having exactly the same convergence rate as the famous 'pretty-when-you-write-it-down' infinite series representation he had copy-pasted, so there's no unique 'slowest-to-converge'
worth noting that for any real number there are an infinite number of infinite sequences that converge to that number. can provide proof if need be, but ya ought to be able to prove that yourself before you go around trying to insult or impress people with oblique references to math.
AMA prints all kinds of things from undergraduate & graduate math students. heck i've been published in it myself, years ago, proving that mathematicians are inherently unfair because the optimal way to slice a pizza for 7 hungry people to share is to actually only slice it into only 6 pieces. so maybe you can see that they don't always have the best rationale or certainly the most accurate article titles.
AMA articles are not above reproach just by virtue of being in an AMA printing --
@cv5 you ought to discard your partiality; according to the Bible it makes you an unjust judge ((James 2:4)). if you want to prove that there is an unique 'most irrational number' you need to do the math. appeals to celebrity or notoriety are worthless when it comes to the establishing of the truth.