The Carathéodory—Fejér Extension of a Finite Geometric Series

It is shown that the Caratheodory—Fejer extension of a finite geometric series can be given explicitly up to a simple polynomial equation in an auxiliary variable. This result allows us to analyse the Caratheodory-Fejer approximation method in the case where the quotients of successive Maclaurin coefficients of the given function tend to a limi