series to rational order

Question

The function Series requires that the expansion order is an integer, i.e. for the input

Series[x^(1/4)+x^(1/3)+x^(1/2),{x,0,1/3}]

it will give the warning

Series::serlim: "Series order specification 1/3 is not a machine-sized integer."

and return the Series unevaluated.

It seems to me that this is a well defined and easily solvable problem though. In fact, we can write our own series function as follows,

mySeries[expr_, {x_, x0_, n_?IntegerQ}] := Series[expr, {x, x0, n}]
mySeries[expr_, {x_, x0_, n_}] :=  Quiet[
Series[ expr /. x -> x^Denominator[n], {x, x0, Numerator[n]}] /. 
 SeriesData[y_, y0_, list_, nmin_, nmax_, nfrac_] :> 
SeriesData[y, y0, list, nmin, nmax, Denominator[n] nfrac], SeriesData::sdatc]

Now,

mySeries[x^(1/4)+x^(1/3)+x^(1/2),{x,0,1/3}]

gives

SeriesData[x, 0, {1, 1}, 3, 6, 12]

(This is x^(1/4)+x^(1/3)+O(x^(1/2))), as expected.

Why is this not the standard behaviour of Series, is there something subtle I'm missing, or is this a bug?

Also note that SeriesCoefficient works perfectly fine with rational powers.

Answer 1

In general your way may be right, but in the example like yours one can do an easy trick. Let us first replace:

  expr = x^(1/4) + x^(1/3) + x^(1/2) /. x -> z^12 // Simplify[#, z > 0] &

(*  z^3 (1 + z + z^3)  *)

Then apply Series:

expr2 = Series[expr, {z, 0, 4}] // Normal

(*  z^3 + z^4  *)

Then replace back:

    expr2 /. z -> x^(1/12) // Simplify[#, x > 0] &

(*  x^(1/4) + x^(1/3)   *)

Have fun!