Substituting one of these things into a linear differential equation with variable coef cients allows one to  nd a recurrence relation to determine the solution.
The ring of the "formal" type of these things is denoted by an R followed by a double-square-bracketed x. Applying a method that adds a potentially non-integer component to these things yields an indicial equation as part of solving a differential equation with a regular singular point via the method of Frobenius. Substituting one of these things into a linear differential equation with variable coef cients allows one to  nd a recurrence relation to determine the solution. Taking the limit of their consecutive coef cients yields the radius of convergence. Using them to solve a differential equation involves shi0ing the index of summation and recognizing which function's Taylor series they correspond to. For 10 points, what constructs are the summation of sequences with exponentiated variables?
The ring of the "formal" type of these things is denoted by an R followed by a double-square-bracketed x.
power series [prompt on "in nite series"]