Creation of Power Series
This is the easiest way to create a power series. This tells Axiom that x is to be treated as a power series, so funcitons of x are again power series. We didn't say anything about the coefficients of the power series, so the coefficients are general expressions over the integers. This allows us to introduce denominators, symbolic constants, and other variables as needed. Here the coefficents are integers (note that the coefficients are the Fibonacci numbers). This series has coefficients that are rational numbers. When you enter this expression you introduce the symbolic constants sin(1) and cos(1). When you enter the expression the variable a appears in the resulting series expansion. You can also convert an expression into a series expansion. This expression creates the series expansion of 1/log(v) about v=1. For details and more examples see Converting to Power Series You can create power series with more general coefficients. You normally accomplish this via a type declaration, see Declarations. See Functions on Power Series for some warnings about working with declared series. We delcare that y is a one-variable Taylor series (UTS is the abbreviation for UnivariateTaylorSeries in the variable z with FLOAT (that is, floating-point) coefficients, centered about 0. Then, by assignment, we obtain the Taylor expansion of exp(z) with floating-point coefficients. You can also create a power series by giving an explicit formula for the nth coefficient. For details and more examples see Power Series from Formulas To create a series about w=0 whose nth Taylor coefficient is 1/n!, you can evaluate this expression. This is the Taylor expansion of exp(w) at w=0.