ABSTRACT: We consider nonlinear recurrences based on sin(x) and cos(x), which possess attractive fixed points 0 and 0.739... (Dottie’s number). Detailed asymptotics for convergence are found, using a 2013 algorithm due to Mavecha & Laohakosol for sin and a 1960 paper by Thron for cos.
​CITATION (two original articles):​
S. Finch (2024), What do sin(x) and arcsinh(x) have in common?, arXiv:2411.01591. https://doi.org/10.48550/arXiv.2411.01591​
S. Finch (2025), Exercises in Iterational Asymptotics II, arXiv:2501.06065. https://doi.org/10.48550/arXiv.2501.06065
Given an initial value
0<
x
0
≤π
, define
x
n
=sin(
x
n-1
)
for
n≥1
. Clearly
x
n
0
as
n∞
. We wish to describe in detail this monotonic convergence.
It is not difficult to prove that
lim
n∞
1/2
n
x
n
=
3
(see Polyá & Szegö [1] and Newman [2]) but harder to show
lim
n∞
3/2
n

x
n

3
-1
1/2
n
ln(n)=-3/10
.
It is even harder to determine that
lim
n∞
3/2
n

x
n

3
-1
1/2
n
+(3/10)ln(n)
3/2
n
=-C/2
, where
C
is a constant that depends on the choice of
x
0
.
Mavecha & Laohakosol [3], building on work of de Bruijn [4] and Bencherif & Robin [5], formulated an algorithm for finding the asymptotic expansion of
x
n
.
The unknown constant
C
can hence be calculated to arbitrary precision, assuming the degree of the asymptotic expansion is suitably high.
We recommend that readers consult [3, 6] for more explanation of theoretical matters underlying the Mathematica code.
In [6], the parameter
k
was fixed to be
7
for simplicity’s sake. Here we take
k=10
. Readers are invited to try other values of
k
in the spirit of exploration.
At the end of this document, we also discuss iterates of cos, which exhibit oscillatory convergence. It is faster than that for sin; our analysis is easier [7].

Sine

Four initial values
x
0
are studied here:
π/2
,
π/3
,
π/4
, π/6.
In[]:=
h[x0_]:={For[n=1;x=x0,n<=3,n++,x=N[Sin[x],1000];​​If[n>0,Print[{n,N[x,8],N[n^(1/2)x,8]}]]],​​For[n=1;x=x0,n<=300000,n++,x=N[Sin[x],1000];​​If[n>0&&Mod[n,100000]==0,Print[{n,N[x,6],​​N[n^(1/2)x,8],N[n^(3/2)(x/Sqrt[3]-1/n^(1/2))/Log[n],8],​​N[-2n^(3/2)(x/Sqrt[3]-1/n^(1/2)+(3/10)Log[n]/n^(3/2)),8]}]]]}
In[]:=
h[Pi/2];
{1,1.0000000,1.0000000}
{2,0.84147098,1.1900197}
{3,0.74562414,1.2914589}
{100000,0.00547700,1.7319786,-0.36210332,1.4299817}
{200000,0.00387290,1.7320129,-0.35858531,1.4301931}
{300000,0.00316223,1.7320248,-0.35670483,1.4302702}
In[]:=
h[Pi/3];
{1,0.86602540,0.86602540}
{2,0.76175998,1.0772913}
{3,0.69019608,1.1954547}
{100000,0.00547698,1.7319717,-0.39691708,2.2315982}
{200000,0.00387289,1.7320094,-0.39142405,2.2318573}
{300000,0.00316223,1.7320225,-0.38848845,2.2319509}
In[]:=
h[Pi/4];
{1,0.70710678,0.70710678}
{2,0.64963694,0.91872537}
{3,0.60489734,1.0477129}
{100000,0.00547693,1.7319566,-0.47219175,3.9648615}
{200000,0.00387287,1.7320019,-0.46242899,3.9652400}
{300000,0.00316222,1.7320175,-0.45721219,3.9653750}
In[]:=
h[Pi/6];
{1,0.50000000,0.50000000}
{2,0.47942554,0.67801010}
{3,0.46126956,0.79894231}
{100000,0.00547678,1.7319085,-0.71373795,9.5266684}
{200000,0.00387282,1.7319778,-0.69028043,9.5275825}
{300000,0.00316219,1.7320015,-0.67774540,9.5279008}
Convergence to
3
=1.7320508
is faster than convergence to
-3/10
. We wish to improve the
C
-value estimates
1.43,2.23,3.96,9.52
.
​
Let us begin. The parameter
τ
, for example, is the gap between successive powers of
x
in the series for
f(x)
.
In[]:=
Clear[x,n]
In[]:=
k:=10
In[]:=
f[x_]:=Sin[x]
In[]:=
Series[f[x],{x,0,2k+1}]
Out[]=
x-
3
x
6
+
5
x
120
-
7
x
5040
+
9
x
362880
-
11
x
39916800
+
13
x
6227020800
-
15
x
1307674368000
+
17
x
355687428096000
-
19
x
121645100408832000
+
21
x
51090942171709440000
+
22
O[x]
In[]:=
A:=Table[Coefficient[Series[f[x],{x,0,2k+1}],x^(2j+1)],{j,1,k}]
In[]:=
A
Out[]=
-
1
6
,
1
120
,-
1
5040
,
1
362880
,-
1
39916800
,
1
6227020800
,-
1
1307674368000
,
1
355687428096000
,-
1
121645100408832000
,
1
51090942171709440000

In[]:=
τ:=2
In[]:=
λ:=-1/(τA[[1]])
In[]:=
λ
Out[]=
3
We need to sum over all nonnegative integers
n
1
,
n
2
,...,
n
k
such that both
n
1
+2
n
2
+3
n
3
+...+k
n
k
=m
and
n
1
+
n
2
+
n
3
+...+
n
k
=m-s
are satisfied.
In[]:=
loopIndices[k_Integer,m_Integer,s_Integer,indx_]/;m≥s≥0&&k>0:=Module[{top=m-s,indices},indices=Table[{indx[j],0,With[{priors=Table[indx[i],{i,j-1}]},Min[top-Total[priors],Floor[(m-Range[j-1].priors)/j]]]},{j,k}];​​indices[[-1,2]]=top-Sum[indx[i],{i,k-1}];​​Apply[Sequence,indices]]
In[]:=
L[k_,m_,s_]:=Flatten[Table[With[{indices=Table[a[j],{j,k}]},If[indices.Range[k]m,indices,Nothing]],Evaluate@loopIndices[k,m,s,a]],k-1]
We also need a power function that returns
0
0
as
1
; no intervention is required when the base is a nonconstant polynomial of
X
.
In[]:=
pwr[a_,b_]:=If[a0&&b0&&NumberQ[a],1,a^b]
The challenge here is to generalize both the multinomial coefficient & power function for vectors of arbitrary length
k
, and finally to perform a summation.
In[]:=
pwrLists[l1_List,l2_List]/;Length[l1]Length[l2]:=Apply[Times,Table[pwr[l1[[j]],l2[[j]]],{j,Length[l1]}]]
In[]:=
mlt[L3_List,L4_List]:=Apply[Multinomial,L3]*pwrLists[L4,L3]
In[]:=
sumLists[L5_List,L6_List]:=Apply[Plus,Table[mlt[L5[[j]],L6],{j,Length[L5]}]]
Various sequences of rational numbers are now defined.
In[]:=
b[j_]:=λ^(j+1)Sum[(-1)^(j+1-s)(Pochhammer[τ,j+1-s]/(j+1-s)!)sumLists[L[k,j+1,s],A],{s,0,j}]
In[]:=
Table[b[j],{j,1,k-1}]
Out[]=

3
5
,
2
7
,
3
25
,
18
385
,
1382
79625
,
12
1925
,
32553
14889875
,
87734
116491375
,
1047666
4094715625

In[]:=
B:=Prepend[Table[b[j],{j,1,k-1}],1]
In[]:=
B
Out[]=
1,
3
5
,
2
7
,
3
25
,
18
385
,
1382
79625
,
12
1925
,
32553
14889875
,
87734
116491375
,
1047666
4094715625

In[]:=
a0[j_]:=a0[j]=Sum[((-1)^(j-1-s)/(j-s))sumLists[L[k,j,s],B],{s,0,j-1}]
In[]:=
Table[a0[j],{j,1,k-1}]
Out[]=
1,
1
10
,
2
105
,
3
700
,
2
1925
,
691
2627625
,
12
175175
,
10851
595595000
,
87734
17823180375

In[]:=
a[i_,j_]:=a[i,j]=Sum[(-1)^(j-i-s)(Pochhammer[i,j-i-s]/(j-i-s)!)sumLists[L[k,j-i,s],B],{s,0,j-i-1}]
In[]:=
c[i_]:=c[i]=(1/i)(b[i+1]-a0[i+1]b[1]+Sum[a[z,i+1]c[z],{z,1,i-1}])
In[]:=
Table[c[i],{i,1,k-2}]
Out[]=

79
350
,
87
875
,
91543
1347500
,
18222899
350350000
,
88627739
2358125000
,
3899439883
195429609375
,-
32544553328689
53370523456250000
,-
36938322475101
2132688250000000

Define polynomials
T
1
(X),
T
2
(X),...
and
P
1
(X),
P
2
(X),...
using
b
and
c
sequences.
In[]:=
t[1,X_]:=X
In[]:=
t[2,X_]:=b[1]X-c[1]
In[]:=
T:=Table[t[j,X],{j,1,2}]
In[]:=
t[m_,X_]:=Expand[b[1]Sum[((-1)^(m-2-s)/(m-1-s))sumLists[L[m-1,m-1,s],T],{s,0,m-2}]-c[m-1]-Sum[c[p]Sum[(-1)^q(Pochhammer[p,q]/q!)sumLists[L[m-1,m-p-1,m-p-q-1],T],{q,1,m-1-p}],{p,1,m-2}]]
In[]:=
For[m=3,m≤k-1,m++,S=T;S=Append[S,t[m,X]];T=S;T]
In[]:=
T[[2]]
Out[]=
-
79
350
+
3X
5
In[]:=
T[[3]]
Out[]=
-
411
1750
+
41X
70
-
3
2
X
10
In[]:=
T[[4]]
Out[]=
-
87519
336875
+
1437X
1750
-
134
2
X
175
+
3
X
5
In[]:=
T[[5]]
Out[]=
-
4499489
14014000
+
857061X
673750
-
5919
2
X
3500
+
31
3
X
35
-
3
4
X
20
In[]:=
T[[6]]
Out[]=
-
26049754659
61311250000
+
16303453X
7962500
-
4795533
2
X
1347500
+
2438
3
X
875
-
683
4
X
700
+
3
5
X
25
In[]:=
S=T;T=(S/.X->-τX);
In[]:=
P[m_,X_]:=If[m0,1,Sum[((-1)^(m-s)Pochhammer[1/τ,m-s]/(m-s)!)sumLists[L[k-1,m,s],T],{s,0,m-1}]]
In[]:=
Expand[P[2,X]]
Out[]=
79
700
+
3X
5
+
3
2
X
2

Cosine

To prove rigorously that the limit exists, we demonstrate in [7] that the following infinite product converges (to the same value).
Again, the limit exists [7, 10] because the following infinite product converges.

Acknowledgement

References

1. G. Pólya and G. Szegő, Problem 173, Problems and Theorems in Analysis. I, Springer-Verlag, 1998, pp. 38, 217; MR1492447.
2. D. J. Newman, Problem 63, A Problem Seminar, Springer-Verlag, 1982, pp. 12, 34, 80--81; MR0678095.
3. S. Mavecha and V. Laohakosol, Asymptotic expansions of iterates of some classical functions, Appl. Math. E-Notes 13 (2013) 77--91; MR3121616; http://www.emis.de/journals/AMEN/2013/2013.htm.
4. N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland, 1958; MR0099564 / Dover, 1981; MR0671583.
7. S. Finch, Exercises in iterational asymptotics II, http://arxiv.org/abs/2501.06065.
8. S. R. Kaplan, The Dottie number, Math. Mag. 80 (2007) 73--74.
9. J.-C. Pain, An exact series expansion for the Dottie number, http://arxiv.org/abs/2303.17962.
10. W. J. Thron, Sequences generated by iteration, Trans. Amer. Math. Soc. 96 (1960) 38--53; MR0117462.

CITE THIS NOTEBOOK

Iterational asymptotics of sine and cosine​
by Steven Finch​
Wolfram Community, STAFF PICKS, January 21, 2025
​https://community.wolfram.com/groups/-/m/t/3362137