f[n_] := If[n == 1, 1, Sqrt[3 f[n - 1] - 1]]
Limit[f[n], n -> Infinity]
Evaluating these two lines makes the Mathematica 8.0 kernel crash.
duplicated here with 8.0.4 and 9.0.1 on Macintosh.
In[1]:= f[n_] := If[n == 1, 1, Sqrt[3 f[n - 1] - 1]]
In[2]:= Limit[f[n], n -> Infinity]
Segmentation fault
A dubious input, but the Kernel should not crash.
I don’t get a segmentation fault on a Mac and Mathematica 9.0.1, but the limit does not evaluate either.
I can, however, evaluate for example
f[n_] := Sqrt[3 f[n - 1] - 1]
f[1] = 1;
N[f[100], 20]
and then use
WolframAlpha[ToString[N[f[100], 20]],
IncludePods -> "PossibleClosedForm"]
That gives the result.
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I used the Mathematica Kernel directly, so the error message showed.
In the Mathematica FrontEnd, the message does not come through.
If you do this right, it will not crash
In[22]:= Remove[weitzerF]
weitzerF[n_?Positive] := weitzerF[n] = Sqrt[3 weitzerF[n - 1] - 1]
weitzerF[1] = 1;
In[30]:= Limit[weitzerF[o], o -> \[Infinity]]
Out[30]= Limit[weitzerF[o], o -> \[Infinity]]
In[31]:= Limit[N[weitzerF[o]], o -> \[Infinity]]
Out[31]= Limit[weitzerF[o], o -> \[Infinity]]
In[32]:= $Version
Out[32]= "9.0 for Microsoft Windows (64-bit) (January 25, 2013)"
but it does also not deliver a result. Therefore, if you do it even more right, you see that the very recursive nature of the weitzerF[] function tends to the usage of FixedPoint[]
In[43]:= weitzerF[1]
Out[43]= 1
In[44]:= weitzerF[2]
Out[44]= Sqrt[2]
In[45]:= weitzerF[3]
Out[45]= Sqrt[-1 + 3 Sqrt[2]]
In[46]:= weitzerF[4]
Out[46]= Sqrt[-1 + 3 Sqrt[-1 + 3 Sqrt[2]]]
and here is the limit
In[53]:= FixedPoint[N[Sqrt[-1 + 3 #], 53] &, Sqrt[-1 + 3 Sqrt[2]]]
Out[53]= 2.6180339887498948482045868343656381177203091798057628
Your original construction was misleading the kernel: The If[] contains a recursive function while n tends to infinity while nothing evaluates to a numerical value! Bad luck. Nonetheless the kernel should not crash and with newer versions it doesn’t.
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You can get candidates for the limit simply by solving an equation for it.
In[83]:= candidates = limF /. Solve[limF == Sqrt[3*limF - 1], limF]
Out[83]= {1/2 (3 - Sqrt[5]), 1/2 (3 + Sqrt[5])}
The crash has been fixed for a future release. It will instead return unevaluated, as it should.
2 Likes
I think the main problem is, is that Limit will not go exactly over the integers. So it will never stop by the f[1] definition, because it might be f[1.56] or so that will be evaluated.
The answer of Daniel is much more practical, and the correct way of solving this in my view.
In[18]:= RootApproximant[FixedPoint[N[Sqrt[-1 + 3 #], 53] &, Sqrt[-1 + 3 Sqrt[2]]], 2]
Out[18]= 1/2 (3 + Sqrt[5])
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You can get candidates for the limit simply by solving an equation for it.
In[83]:= candidates = limF /. Solve[limF == Sqrt[3*limF - 1], limF]
Out\[83]= {1/2 (3 - Sqrt\[5]), 1/2 (3 + Sqrt\[5])}
One might ask what the 1/2 (3 - Sqrt[5]) candidate is for. In a way it belongs to the reverse directed weitzerF[], if on writes
weitzerF[n-1] (* arrrgh *) := Sqrt[3 weitzerF[n] - 1]
or with other words
In[1]:= Remove[uF]
uF[n_Integer?Positive] := uF[n] = (uF[n - 1]^2 + 1)/3
uF[1] := 1
In[4]:= uF[1]
Out[4]= 1
In[5]:= uF[2]
Out[5]= 2/3
In[6]:= uF[3]
Out[6]= 13/27
In[7]:= uF[4]
Out[7]= 898/2187
In[9]:= uF[20] // N
Out[9]= 0.381966
In[14]:= RootApproximant[N[uF[25]], 2]
Out[14]= 1/2 (3 - Sqrt[5])
Mma 9.0.1. does not evaluate uF[30] in an useful span of time - because rational numbers with very big numerators and denominators are involved.
Here are the cob-web-plots for this function:
As you see there are two solutions:
1/2 (3 \[PlusMinus] Sqrt[5])
One is stable, one is unstable, as you can see from the cob-web-plot above.
Play around with it yourself using this code:
ClearAll[CobwebPlot]
SetAttributes[CobwebPlot,HoldAll]
CobwebPlot[f_,start_?NumericQ,n_,{xrange:{xmin_,xmax_},yrange:{_,_}}]:=Module[{cob,x,g1,coor},
cob=NestList[f,start,n];
coor = Partition[Riffle[cob,cob],2,1];
coor[[1,2]]=0;
g1=Graphics[{Red,Line[coor]}];
Show[{Plot[{x,f[x]},{x,xmin,xmax},PlotStyle->{{Thick,Black},Black },PlotRange->{xrange,yrange}],g1}]
]
Manipulate[CobwebPlot[Sqrt[3 #-1]&,\[Beta],50,{{0,4},{0,4}}],{{\[Beta],1},0,5}]
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One is stable, one is unstable, as you can see from the cob-web-plot above.
You mean 1/2 (3 + Sqrt[5]) is stable because both cob-web-plots reach it?
And 1/2 (3 - Sqrt[5]) is unstable because no cob-web-plot reaches it?
But you can easily reach the 1/2 (3 - Sqrt[5]) point by interchanging the role of the x = y line with the role of the function f: Starting at ?, stop first on the y = x line, turn left to the function, go down to the y = x line, turn left to the function and so on … This modified procedure will not turn to an end if ? > 1/2 (3 + Sqrt[5]) because 1/2 (3 + Sqrt[5]) is the biggest crossing point between the y = x line and the function. So here is the orthodox cob-web-plot with the uF function:
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