In[]:=
a=2;Solve[Tan[x]==ax,x]
​
Solve
:This system cannot be solved with the methods available to Solve.
Out[]=
Solve[Tan[x]2x,x]
In[]:=
Series[HarmonicNumber[m,3]+PolyGamma[2,1]/2+1/2/(m+1/2+(1/8)/(m+1/2+(8*19/3/128)/(m+1/2+a/(m+1/2))))^2,{m,
Infinity,10}]
Out[]=
713
18432
-
19a
384
m
8
+
-
713
4608
+
19a
96
m
9
+
19(25999-45120a+5760a
2
)
2211840m
10
+O
1
m

11
In[]:=
Series[1/(2m)/(1/m/(Pi^2/6-HarmonicNumber[m,2])-1)-1,{m,Infinity,10}]
Out[]=
1-
1
6m
+
1
9m
2
-
4
135m
3
-
13
405m
4
+
251
8505m
5
+
3532
127575m
6
-
19139
382725m
7
-
48533
1148175m
8
+
23992492
189448875m
9
+
395591041
3978426375m
10
+O
1
m

11
In[]:=
t=1/(6m)/(1/(2m)/(1/m/(Pi^2/6-HarmonicNumber[m,2])-1)-1);​
​Series[t,{m,Infinity,10}]
Out[]=
-1-
2
3m
-
4
15m
2
+
2
15m
3
+
46
525m
4
-
104
525m
5
-
82
2625m
6
+
226
525m
7
-
143932
1010625m
8
-
182134
144375m
9
+
940406
853125m
10
+O
1
m

11
In[]:=
Series[HarmonicNumber[m,2]-Pi^2/6,{m,Infinity,5}]
Out[]=
-
1
m
+
1
2m
2
-
1
6m
3
+
1
30m
5
+O
1
m

6
In[]:=
Integrate[1/(2x)-1/(xE^x),{x,a,k/a},Assumptions>0]
Out[]=
1
2
(-2ExpIntegralEi[-x]+Log[x])
In[]:=
​
​
Syntax
:"Pi^" cannot be followed by "/6".
In[]:=
PadeApproximant[1/2/(HarmonicNumber[m,3]+1/2PolyGamma[2,1])-m-1/2,{m,∞,5}]//Apart
Out[]=
-1-2m-m
2
+
1
6(3+2m+2m
2
)
In[]:=
Series[(1/m/(Pi^2/6-HarmonicNumber[m,2])),{m,∞,5}]
Out[]=
1+
1
2m
+
1
12m
2
-
1
24m
3
-
1
720m
4
+
11
480m
5
+O
1
m

6
In[]:=
Series[Pi^2/6-HarmonicNumber[m,2],{m,∞,5}]
Out[]=
1
m
-
1
2m
2
+
1
6m
3
-
1
30m
5
+O
1
m

6
In[]:=
MinimalPolynomial[Sin[Pi/90],x]//Factor
Out[]=
1-24x-144x
2
+248x
3
+1680x
4
-864x
5
-7168x
6
+1152x
7
+13824x
8
-512x
9
-12288x
10
+4096x
12
In[]:=
MinimalPolynomial[Sin[Pi/36],x]//Factor
Out[]=
1-144x
2
+1680x
4
-7168x
6
+13824x
8
-12288x
10
+4096x
12
In[]:=
ChebyshevT[12,x]
Out[]=
1-72x
2
+840x
4
-3584x
6
+6912x
8
-6144x
10
+2048x
12
In[]:=
Print["Mathematica:",$Version," time unit=",$TimeUnit];​
​​
​digArr={20,30,40,50,60,70,80,90,100,125,158,199,251,316,398,501,630,794,​
​  1000};​
​​
​For[i=1,i≤Length[digArr],i++,​
​  prec=digArr[[i]];​
​  n=IntegerPart[N[2^24/(prec^1.5),10]]; ​
​  n=Max[n,3];​
​  (*Print["n=",n];*);​
​  x=N[Sqrt[3]-1,prec];​
​  y=N[Sqrt[5],prec];​
​  ​
​  While[True, ​
​        (*t=(AbsoluteTiming[Do[Log[x],{n}]][[1]]-AbsoluteTiming[Do[Null,{n}]][[1]]);*)​
​    t=Timing[Do[Log[x],{n}]][[1]];​
​    (* convert to ms *)​
​    t*=1000; ​
​    (*Print["t=",t];    *)​
​    If[t≥10.0,Break[]];​
​    n*=2;​
​  ];​
​  Print[prec,":took ",t/n," ms (",n," eval in ",t/15.625,"ms)"];​
​]​
​​
​​
​
Mathematica:11.2.0 for Linux x86 (64-bit) (September 11, 2017) time unit=
1
100
20:took 0.00112472 ms (187574 eval in 13.502ms)
30:took 0.00133181 ms (102102 eval in 8.70272ms)
40:took 0.00135691 ms (66317 eval in 5.7591ms)
50:took 0.00130637 ms (47453 eval in 3.96742ms)
60:took 0.00135719 ms (36098 eval in 3.13549ms)
70:took 0.00150087 ms (28646 eval in 2.75162ms)
80:took 0.00157784 ms (23446 eval in 2.36762ms)
90:took 0.00173011 ms (19649 eval in 2.17568ms)
100:took 0.00178786 ms (16777 eval in 1.91968ms)
125:took 0.00191578 ms (12004 eval in 1.47181ms)
158:took 0.00236735 ms (8447 eval in 1.27981ms)
199:took 0.00284421 ms (5976 eval in 1.08781ms)
251:took 0.00331863 ms (4218 eval in 0.895872ms)
316:took 0.00401808 ms (2986 eval in 0.767872ms)
398:took 0.00615436 ms (2112 eval in 0.831872ms)
501:took 0.00935695 ms (1496 eval in 0.895872ms)
630:took 0.0113189 ms (1060 eval in 0.767872ms)
794:took 0.0186889 ms (749 eval in 0.895872ms)
1000:took 0.0264113 ms (530 eval in 0.895872ms)