Power Series

The mathematical operations we have discussed so far are exact. Given precise input, their results are exact formulas.

In many situations, however, you do not need an exact result. It may be quite sufficient, for example, to find an approximate formula that is valid, say, when the quantity is small.

This gives a power series approximation to for close to , up to terms of order .
In[1]:=
Click for copyable input
Out[1]=
Mathematica knows the power series expansions for many mathematical functions.
In[2]:=
Click for copyable input
Out[2]=
If you give it a function that it does not know, Series writes out the power series in terms of derivatives.
In[3]:=
Click for copyable input
Out[3]=

Power series are approximate formulas that play much the same role with respect to algebraic expressions as approximate numbers play with respect to numerical expressions. Mathematica allows you to perform operations on power series, in all cases maintaining the appropriate order or "degree of precision" for the resulting power series.

Here is a simple power series, accurate to order .
In[4]:=
Click for copyable input
Out[4]=
When you do operations on a power series, the result is computed only to the appropriate order in .
In[5]:=
Click for copyable input
Out[5]=
This turns the power series back into an ordinary expression.
In[6]:=
Click for copyable input
Out[6]=
Now the square is computed exactly.
In[7]:=
Click for copyable input
Out[7]=
Applying Expand gives a result with 11 terms.
In[8]:=
Click for copyable input
Out[8]=
Series[expr,{x,x0,n}]find the power series expansion of expr about the point to at most n^(th) order
Normal[series]truncate a power series to give an ordinary expression

Power series operations.

New to Mathematica? Find your learning path »
Have a question? Ask support »