Basic Examples

This is a power series for the exponential function around
x0
:

○
NSeries
[Exp[x],{x,0,5}]
In[]:=
-
5.78185×10
-17
+4.36677×10
-18

x
5
-
6.245×10
-17
-8.03265×10
-18

x
4
-
1.38778×10
-16
-4.57183×10
-17

x
3
-
1.09938×10
-16
+2.87761×10
-18

x
2
-
1.85941×10
-17
+4.45741×10
-17

x
+(1.-4.54297×10
-17
)+1.-1.67207×10
-16
x+0.5-1.49215×10
-16
x
2
+(0.166667-7.22073×10
-17
)x
3
+(0.0416667-2.82499×10
-17
)x
4
+(0.00833333-2.93059×10
-17
)x
5
+O[x]
6
Out[]=

Chop
is needed to eliminate spurious residuals:

%//Chop
In[]:=
1.+x+0.5x
2
+0.166667x
3
+0.0416667x
4
+0.00833333x
5
+O[x]
6
Out[]=

Using extended precision may also eliminate spurious imaginaries:

○
NSeries
[Exp[x],{x,0,5},WorkingPrecision->17]
In[]:=
1.00000000000000+x+0.500000000000000x
2
+0.166666666666667x
3
+0.041666666666667x
4
+0.008333333333333x
5
+O[x]
6
Out[]=

Scope

In[]:=

Find expansions in the complex plane:

○
NSeries
[Exp[x],{x,I,5}]//Chop
In[]:=
(0.540302+0.841471)+(0.540302+0.841471)(x-)+(0.270151+0.420735)(x-)
2
+(0.0900504+0.140245)(x-)
3
+(0.0225126+0.0350613)(x-)
4
+(0.00450252+0.00701226)(x-)
5
+O[x-]
6
Out[]=

Find Laurent expansions about essential singularities:

○
NSeries
[Sin[x+1/x],{x,0,10}]//Chop
In[]:=
2.49234×10
-6
x
9
-
0.000174944
x
7
+
0.00703963
x
5
-
0.128943
x
3
+
0.576725
x
+0.576725x-0.128943x
3
+0.00703963x
5
-0.000174944x
7
+2.49234×10
-6
x
9
+O[x]
11
Out[]=

Series
will not find Laurent expansions about essential singularities:

Series[Sin[x+1/x],{x,0,10}]
In[]:=
Sin
1
x
+x+O[x]
11
Out[]=

Options

In[]:=

Radius

Use
"Radius"
to pick the annulus within which the Laurent series will converge:

○
NSeries
1
(1+x)(3+x)
,{x,0,10},"Radius"->2//Chop
In[]:=
-
0.5
x
10
+
0.5
x
9
-
0.5
x
8
+
0.5
x
7
-
0.5
x
6
+
0.5
x
5
-
0.5
x
4
+
0.5
x
3
-
0.5
x
2
+
0.5
x
-0.166667+0.0555556x-0.0185185x
2
+0.00617284x
3
-0.00205761x
4
+0.000685871x
5
-0.000228624x
6
+0.0000762079x
7
-0.0000254026x
8
+8.46754×10
-6
x
9
-2.82251×10
-6
x
10
+O[x]
11
Out[]=

Laurent series for
x≥3
:

○
NSeries
1
(1+x)(3+x)
,{x,0,10},"Radius"->5//Chop
In[]:=
9841.
x
10
-
3280.
x
9
+
1093.
x
8
-
364.
x
7
+
121.
x
6
-
40.
x
5
+
13.
x
4
-
4.
x
3
+
1
x
2
+O[x]
11
Out[]=

Changing
"Radius"
can improve accuracy:

○
NSeries
1
1-x
,{x,0,10},"Radius"->
7
8
//Chop
In[]:=
0.000194357
x
10
+
0.000194357
x
9
+
0.000194357
x
8
+
0.000194357
x
7
+
0.000194357
x
6
+
0.000194357
x
5
+
0.000194357
x
4
+
0.000194357
x
3
+
0.000194357
x
2
+
0.000194357
x
+1.00019+1.00019x+1.00019x
2
+1.00019x
3
+1.00019x
4
+1.00019x
5
+1.00019x
6
+1.00019x
7
+1.00019x
8
+1.00019x
9
+1.00019x
10
+O[x]
11
Out[]=
○
NSeries
1
1-x
,{x,0,10},"Radius"->
1
5
//Chop
In[]:=
1.+x+x
2
+x
3
+1.x
4
+1.x
5
+1.x
6
+1.x
7
+1.x
8
+1.x
9
+1.x
10
+O[x]
11
Out[]=

Applications

A function defined only for numerical input:

f[a_?NumericQ]:=NIntegrate[Sin[aSin[x]+Cos[x]],{x,0,a}]
In[]:=

Find a series expansion of
f
:

g[a_]=Normal@
○
NSeries
[f[a],{a,0,10}]//Chop
In[]:=
0.841471a+0.180101a
3
-0.0741081a
5
+0.00852025a
7
+0.00134129a
9
Out[]=

Check:

Plot[{f[a],g[a]},{a,0,2}]
In[]:=
Out[]=

Properties & Relations

Possible Issues

In[]:=

The correct expansion is analytic at the origin:

Neat Examples

In[]:=

Find the series expansion of the generating function for unrestricted partitions:

Check: