Expansion Point

analysis

Definition

Expansion Point

In the context of a power series or a Taylor series, the expansion point (or development point) is the constant $x_0$ around which the series is centred. A power series with expansion point $x_0$ is defined as:

$$f(x)=\sum _{n=0}^{\infty }{a}_n(x-x_0{)}^n$$

For a Taylor series of a function $f$, the expansion point is the value at which the function’s derivatives are evaluated to determine the coefficients $a_n$.

Properties

• Exactness: At the expansion point $x=x_0$, the power series always converges, and its value is exactly $a_0=f(x_0)$.
• Radius of Convergence: The series represents the function within a symmetric interval $(x_0-R,x_0+R)$.
• Local Approximation: The Taylor polynomial provides the “best” local polynomial approximation of the function near the expansion point.