MATHEMATICA TUTORIAL

Operations on Power Series

Mathematica allows you to perform many operations on power series. In all cases, Mathematica gives results only to as many terms as can be justified from the accuracy of your input.

Here is a power series accurate to fourth order in .
In[1]:=
Click for copyable input
Out[1]=
When you square the power series, you get another power series, also accurate to fourth order.
In[2]:=
Click for copyable input
Out[2]=
Taking the logarithm gives you the result , but only to order .
In[3]:=
Click for copyable input
Out[3]=

Mathematica keeps track of the orders of power series in much the same way as it keeps track of the precision of approximate real numbers. Just as with numerical calculations, there are operations on power series which can increase, or decrease, the precision (or order) of your results.

Here is a power series accurate to order .
In[4]:=
Click for copyable input
Out[4]=
This gives a power series that is accurate only to order .
In[5]:=
Click for copyable input
Out[5]=

Mathematica also allows you to do calculus with power series.

Here is a power series for .
In[6]:=
Click for copyable input
Out[6]=
Here is its derivative with respect to .
In[7]:=
Click for copyable input
Out[7]=
Integrating with respect to gives back the original power series.
In[8]:=
Click for copyable input
Out[8]=

When you perform an operation that involves both a normal expression and a power series, Mathematica "absorbs" the normal expression into the power series whenever possible.

The is automatically absorbed into the power series.
In[9]:=
Click for copyable input
Out[9]=
The is also absorbed into the power series.
In[10]:=
Click for copyable input
Out[10]=
If you add Sin[x], Mathematica generates the appropriate power series for Sin[x], and combines it with the power series you have.
In[11]:=
Click for copyable input
Out[11]=
Mathematica also absorbs expressions that multiply power series. The symbol is assumed to be independent of .
In[12]:=
Click for copyable input
Out[12]=

Mathematica knows how to apply a wide variety of functions to power series. However, if you apply an arbitrary function to a power series, it is impossible for Mathematica to give you anything but a symbolic result.

Mathematica does not know how to apply the function to a power series, so it just leaves the symbolic result.
In[13]:=
Click for copyable input
Out[13]=
New to Mathematica? Find your learning path »
Have a question? Ask support »