The introduction of the uniqueness of function power series expansion in advanced mathematics textbooks is shown in the following figure. There is also a proof process in the textbook. The proof method is that the hypothesis is not unique, and subtracting zero will lead to contradiction, so it is unique. There are also examples in the textbook. The proof process and examples are too complicated to type. If necessary, please check the textbook yourself.

Extended data:

Definition of power series:

Let U 1 (x), U2 (x), ..., UN (x), ... be function columns be defined on the interval I, then the expression

( 1)

It is cal that series of function term defined on the interval I. ..

If each un(x) in the formula (1) is a power function defined on the interval (-∞,+∞), then the function term is a series.

(2)

It is called a power series, where x0 is a constant, a0, a 1, ..., and a coefficient called a power series.