Their basic idea is to extract hidden information contained in partial sums of a specific slowly

convergent or divergent series, and to use that information in order to make a qualified estimate

about new (usually higher order) partial sums which eventually converge to some limit. In many

cases, this “qualified estimate” leads to spectacular numerical results which represent a drastic

improvement over a term-by-term summation of the original series, even if the series is formally

convergent. For further discussion it is useful to consider a sequence {{sn}} = {{s0, s1, . . .}}

with elements sn or the terms an = sn − sn−1 of an infinite series. Sequence transformations are

important tools for the convergence acceleration of slowly convergent sequences or series and

also for the summation of divergent series. The basic idea is to construct from a given sequence

{{sn}} a new sequence {{s′n}} = T({{sn}}) where each s′n depends on a finite number of ele

ments sn1, . . . , snm. Often, the sn are the partial sums of an infinite series. The aim is to find a

transformation T such that {{s′n}} converges faster than sn or, after all, it is capable to sum

{{sn}}. A common approach is to rewrite sn as sn = s + Rn (1) where s is the limit (or anti-limit

in the case of divergence) and Rn is the remainder. The aim then is to find a new sequence {{s′n}}

such that s′n = s + R′n , R′n /Rn → 0 for n → ∞ (2) Thus, the sequence {{s′n}} converges faster

to the limit s (or diverges less violently) than {{sn}}.

## Wednesday, December 1, 2010

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Yeargh. Convergent series. Serious Emergency! Triceratops.

ReplyDeletei love that a slovak wrote this text! (i'm half slovak)

ReplyDeleteI know I am mostly just dumb, but I saw in this some of that old dream of changing linguistics in math, but even moreso, an idea of what it might be like to be able to do computational chemistry.

ReplyDeleteThink of something like 'series equilibrium' in redox reactions within the cell. that's real. redox or oxidation reduction reactions halt themthelves and leave partial byproducts, and more importantly leave like a balanced spectrum of half-redox'd a quarter redoxed. its like a combination of

calculation and process

or maybe even like someone

who stores their own chewed food

like

i half chewed this

i'll hold it

i quarter chewd this

i'll hold it

i 1/8 chewed this

etc

Chris! man i miss you buddy

wish we could hang out

and watch a bluray or something!

:)

Thanks for the support PG!

3 old NW'ers chillin.

your comment reminds me a bit of mechanical linkages and how they might be used in an analytic engine-type machine, except in an electron-shell type arrangement at the atomic level. kind of like the way things are connected together through hydraulics sometimes, where the pressure differentials "solve" the equation of equilibrium. have you seen those charts of the dumb-bell looking electron shell shapes? i absolutely love that imagery. this brings to mind something that i remember reading about that was discovered during the research for the manhatten project, about the specific viscosity of the atomic shells. ??? can't think of what it was right now. would really dig kicking back with a glass of microbrew, a puff of the kind and a trippy sci-fi movie. got to get out that way sometime. i haven't been to texas in almost 30 years!

ReplyDeleteyeeah. like dat.

ReplyDelete:)

we got a new laptop!

the old one died.

have you done any shows lately?

give me a call sometime!

lq