1. Basic theory: The basic theory of power series expansion is very mature, including convergence, convergence radius and uniqueness of power series. The most famous ones are Weierstrassm-test and Abel theorem.

2. Application field: Power series expansion is widely used in various mathematical and physical problems. For example, in calculus, power series can be used to represent Taylor series of functions, so as to make approximate calculation and analysis. In mathematical physics, power series expansion is also used to describe the behavior of physical systems. For example, in quantum field theory and statistical physics, power series expansion is used to calculate various physical quantities.

3. Research in recent years: In recent years, the research focus of power series expansion has shifted from basic theory to more application problems, such as power series expansion of multiple complex variables, power series representation of special functions, power series method in nonlinear partial differential equations and so on.

In addition, the calculation method of power series expansion has also been greatly improved, such as adaptive grid method, boundary element method and other numerical methods have been widely used in power series expansion calculation.

4. Unsolved problems: Although the basic theory of power series expansion is mature, there are still many interesting problems to be solved. For example, how to expand non-analytic functions with power series, how to apply power series expansion to high-dimensional problems, how to combine power series expansion with other mathematical methods and so on. The solution of these problems will promote the development of power series expansion research in a deeper direction.

Power series is one of the important concepts in mathematical analysis, which means that every term in the series is a power of (x-a), which is a constant multiple of the series number n (n is an integer counting from 0, and a is a constant). Power series is an important concept in mathematical analysis, which has been used as a basic content in many fields such as real variable function and complex variable function.

Power series solution is a method to solve ordinary differential equations, especially when the solution of differential equations cannot be expressed by elementary functions or their integrals, other solutions, especially approximate solutions, are needed. Power series solution is a common approximate solution. Many important ordinary differential equations in mathematics and physics can be solved by power series solutions and generalized power series solutions, such as Bessel equation and Legendre equation.