Construction of rational and negative powers of a formal series

Some methods are described for the generation of fractional and negative powers of any formal series, such as Poisson series or Chebyshev series. It is shown that, with the use of the three elementary operations of addition, subtraction, and multiplication, all rational (positive and negative) powers of a series can be constructed. There are basically two approaches: the binomial theorem and the iteration methods. Both methods are described here, and the relationship between them is pointed out. Some well-known classical formulas are obtained as particular cases, and it is shown how the convergence properties of these formulas can be improved with very little additional computations. Finally, at the end of the article, some numerical experiments are described with Chebyshev series and with Fourier series.