- #36
Borek
Mentor
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CRGreathouse said:[tex]R=\sum_{k=1}^\infty10^{-k}r_k[/tex] with [itex]r_1=3,r_2=3,r_3=5,r_4=4,r_5=1,r_6=8,r_7=0,r_8=3,\ldots.[/itex]
:rofl:
Apples and oranges.
CRGreathouse said:[tex]R=\sum_{k=1}^\infty10^{-k}r_k[/tex] with [itex]r_1=3,r_2=3,r_3=5,r_4=4,r_5=1,r_6=8,r_7=0,r_8=3,\ldots.[/itex]
Kurret said:Which shows that you misunderstood my post, since it looks like you have just solved the equation with a calculator and defined a sequence {rk} being the decimal expansion, right?
My question was, if you can find a solution on the form
[tex]\sum_{k=1}^{\infty} f(k,x)[/tex] where f is a function composed of elementary functions (or better, only polynomials), or maybe including more airthmetic/geometric sums of elementary functions.
Im sorry but i don't know how I can define it better. The sum posted is not a polynomial, its a sum of an already defined sequence of numbers, the terms in the sequence isn't a function.Borek said:Arrgh, I can't edit anything, even quoting doesn't work now for me :( Neither in Opera nor in IE.
Kurret: you must concentrate on defining EXACTLY what you mean. I believe most people here understand what you are aiming at, but so far they had no problems with showing that intuition fails here. As long as you will not precisely define what is allowed in the expression you are looking for, you are on the lost position.
qspeechc said:How do you generate the [tex]r_n[/tex]'s?
Kurret said:My question was, if you can find a solution on the form
[tex]\sum_{k=1}^{\infty} f(k,x)[/tex] where f is a function composed of elementary functions (or better, only polynomials), or maybe including more airthmetic/geometric sums of elementary functions.
I don't get thisCRGreathouse said:But they are. f(1, x) is a very simple combination of elementary functions [itex]3\cdot2^k[/itex].
Kurret said:I don't get this
All terms should be the same function, right? Or do I completely misunderstand what you mean?
Yes I agree on that. You showed a good example of why it isn't possible with polynomials, so that question is now answered. So now, let's consider all elementary functions, which that clearly is (the factorial is a geometric sum of the elementary function f(x)=x).DavidWhitbeck said:Kurret that is not an example of what you said though-- the terms are not a polynomial in n. If you want the coefficients in your series to be represented by any map from N to R then certainly the decimal expansion is just as acceptable.
I think it's time for you to rephrase what you mean. I'm having a hard time understanding what you're asking since you seem to be inconsistent about it, and I doubt I'm alone in that regard.
Kurret said:CRgreatHouse, I think that is something like the answer I was looking for :). Nice!
Can you explain a bit furthermore why it does work, ie a non math/geometric explanation? :p can't really understand that atm.
CRGreathouse said:To 500 places:
0.3354180323849400594578624117492548397118513151006308650233458056997707969
93488867326823297805099655941760946007417735305449140296885174971232582467743747
67817725202600726615999809586989494215709825435975337739655562116739626518112856
23055310151600910291959720327473494910708608367675240373868492976882512228069498
07934021595150112613380719098738548853419012578533010207808040687825471704745279
60925128393175327764000215850832461166054705141754704841341722301915669142357206
871814814939461336135943871
>Digits:=500;
Digits := 500
> fsolve(2*x+sin(x)=1);
0.33541803238494005945786241174925483971185131510063086502334580\
5699770796993488867326823297805099655941760946007417735305\
4491402968851749712325824677437476781772520260072661599980\
9586989494215709825435975337739655562116739626518112856230\
5531015160091029195972032747349491070860836767524037386849\
2976882512228069498079340215951501126133807190987385488534\
1901257853301020780804068782547170474527960925128393175327\
7640002158508324611660547051417547048413417223019156691423\
57206871814814939461336135943871
LukeD said:However, in general, no one knows if there is a "nice formula" for your problem. And really, the only reason for that is because either no one has yet figured one out or no one has been trying to find one. I don't know if it's the case that every real number will be able to written in some nice formula though.
CGUE said:Haha win, don't we all love Mathematics Computing Packages?
secant_root(ff, first, second)={
local(ff1, ff2, oldfirst, ep);
ff2 = ff(second);
ep = 2 * eps();
while (abs(first - second) > ep,
ff1 = ff(first);
oldfirst = first;
first = first - (first - second) / (ff1 - ff2) * ff1;
second = oldfirst;
ff2 = ff1;
);
first
}
\\ 2 ^ -(decimal_precision * log(10)/log (2) / 32) * 32 - 1)
eps()={
precision(2. >> (32 * ceil(precision(1.) * 0.103810252965230073370947482171543443)), 9)
}