Definition
Expansion Point
In the context of a power series or a Taylor series, the expansion point (or development point) is the constant
around which the series is centred. A power series with expansion point is defined as: For a Taylor series of a function
, the expansion point is the value at which the function’s derivatives are evaluated to determine the coefficients .
Properties
- Exactness: At the expansion point
, the power series always converges, and its value is exactly . - Radius of Convergence: The series represents the function within a symmetric interval
. - Local Approximation: The Taylor polynomial provides the “best” local polynomial approximation of the function near the expansion point.