In[]:=
ClearAll["`*"];Remove["`*"];
Remove
:There are no symbols matching "`*.
Reduce[ForAll[x,0<x<Pi/2,-Pi<=a*x<=Pi],a,Reals](*Workaround*)
Out[]=
-2≤a≤2
(*Anotherworkaround:*)
In[]:=
S[m_]:=Asymptotic[Sin[ax],{x,0,m}]
In[]:=
S[3]
Out[]=
ax-
3
a
3
x
6
In[]:=
Reduce[ForAll[x,0<x<π/2,2a*(%)>=0],a,Reals]
Out[]=
-
2
6
π
≤a≤
2
6
π
In[]:=
%//N
Out[]=
-1.55939≤a≤1.55939
In[]:=
S[10]
Out[]=
ax-
3
a
3
x
6
+
5
a
5
x
120
-
7
a
7
x
5040
+
9
a
9
x
362880
In[]:=
Reduce[ForAll[x,0<x<π/2,2a*(%)>=0],a,Reals]
Out[]=
-
4
39.7
…
π
≤a≤
4
39.7
…
π
In[]:=
%//N
Out[]=
-2.00452≤a≤2.00452
In[]:=
S[20]
Out[]=
ax-
3
a
3
x
6
+
5
a
5
x
120
-
7
a
7
x
5040
+
9
a
9
x
362880
-
11
a
11
x
39916800
+
13
a
13
x
6227020800
-
15
a
15
x
1307674368000
+
17
a
17
x
355687428096000
-
19
a
19
x
121645100408832000
In[]:=
Reduce[ForAll[x,0<x<π/2,2a*(%)>=0],a,Reals]
Out[]=
-
9
39.5
…
π
≤a≤
9
39.5
…
π
In[]:=
%//N
Out[]=
-2.≤a≤2.