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 x is small.

This gives a power series approximation to (((1+x))^n) for x close to 0, up to terms of order (x^3):

Series[(1+x)^n,{x,0,3}]

The Wolfram Language knows the power series expansions for many mathematical functions:

Series[Exp[-a t] (1+Sin[2 t]),{t,0,4}]

If you give it a function that it does not know, (RefLink[Series, paclet:ref/Series]) writes out the power series in terms of derivatives:

Series[1+f[t],{t,0,3}]

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. The Wolfram Language 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 (x^5):

Series[Exp[x],{x,0,5}]

When you do operations on a power series, the result is computed only to the appropriate order in x:

%^2 (1+%)

This turns the power series back into an ordinary expression:

Normal[%]

Now the square is computed exactly:

%^2

Applying (RefLink[Expand, paclet:ref/Expand]) gives a result with 11 terms:

Expand[%]

Power series operations.