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Here is a rule set that will do the job:

[ a*(b + c) := a*b + a*c, opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m :: constant(a) :: constant(b), opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m :: constant(a) :: constant(b), a O(x^n) := O(x^n) :: constant(a), x^opt(m) O(x^n) := O(x^(n+m)), O(x^n) O(x^m) := O(x^(n+m)) ]

If we really want the `+` and `*` keys to operate naturally
on power series, we should put these rules in `EvalRules`

. For
testing purposes, it is better to put them in a different variable,
say, `O`

, first.

The first rule just expands products of sums so that the rest of the
rules can assume they have an expanded-out polynomial to work with.
Note that this rule does not mention ‘`O`’ at all, so it will
apply to any product-of-sum it encounters—this rule may surprise
you if you put it into `EvalRules`

!

In the second rule, the sum of two O’s is changed to the smaller O.
The optional constant coefficients are there mostly so that
‘`O(x^2) - O(x^3)`’ and ‘`O(x^3) - O(x^2)`’ are handled
as well as ‘`O(x^2) + O(x^3)`’.

The third rule absorbs higher powers of ‘`x`’ into O’s.

The fourth rule says that a constant times a negligible quantity
is still negligible. (This rule will also match ‘`O(x^3) / 4`’,
with ‘`a = 1/4`’.)

The fifth rule rewrites, for example, ‘`x^2 O(x^3)`’ to ‘`O(x^5)`’.
(It is easy to see that if one of these forms is negligible, the other
is, too.) Notice the ‘`x^opt(m)`’ to pick up terms like
‘`x O(x^3)`’. Optional powers will match ‘`x`’ as ‘`x^1`’
but not 1 as ‘`x^0`’. This turns out to be exactly what we want here.

The sixth rule is the corresponding rule for products of two O’s.

Another way to solve this problem would be to create a new “data type”
that represents truncated power series. We might represent these as
function calls ‘`series( coefs, x)`’ where

`series`

objects, and as an optional convenience could also know how to combine a
`series`

object with a normal polynomial. (With this, and with a
rule that rewrites ‘`series`

form,
you could still enter power series in exactly the same notation as
before.) Operations on such objects would probably be more efficient,
although the objects would be a bit harder to read.
Some other symbolic math programs provide a power series data type
similar to this. Mathematica, for example, has an object that looks
like ‘`PowerSeries[ x, x0, coefs, nmin,
nmax, den]`’, where

`PowerSeries`

objects have a special display format that makes them look like
‘