monotonic decrease of sin (revivalist version)

John Conway conway at math.Princeton.EDU
Mon Apr 19 18:47:32 CEST 1999

On Mon, 19 Apr 1999, vdmcc wrote:

> Neil,
> for what it's worth :
> ----------------------------------------------------------------------------
> ---
> Sin[n] decreases monotonically to -1 :

> is a dumb series, because the differences are
> Rest[% ]-Drop[% ,-1]
> {2,1,1,6,333,710,710,710...
> ----------------------------------------------------------------------------
> ----------------------------------------------------------------------------
> ---
> Abs[Cos[n]] decreases monotonically to 0 :

> is again dumb, the differences (of the unsigned arguments) being
> {1,3,3,3,333,355,355,355,355,355...

> So let's be carefull with Gerards early generalisations (;-))

    I think it would be wise to be careful of calling a series
dumb because after a few terms the differences seem to be constant.

   Of course this happens for a while here because 355/113 is such a
good approximation to  pi.  What one obviously gets in all these
cases (possibly after one or two rogue terms) is just the set
of numerators or denominators of the best approximations to
something, including the so-called "intermediate approximants",
which, if there's a big partial quotient, will include a correspondingly
long arithmetical progression.

   Of course it might also be that the program being used to compute
sines (or whatever) isn't accurate enough.  It's probably more sensible
not to compute any trig functions, but just prove the relevant
stuff about approximants.

   John Conway

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