An infinite series of the form
is called a "power series" expansion around the center ζ0 with constant "coefficients" ck. The variable z and the constants ζ0 and ck may be real or complex numbers. For a given z = z0, (1) becomes an infinite series of constant terms cn (z0 − ζ0)n which may or may not be convergent; in the first case, we denote the sum by S(z0). We say that R is the "radius of convergence" of (1) if this series converges for all z with |z − ζ0| < R and diverges for all z with |z − ζ0| > R. Certainly, every series (1) converges at z = ζ0. Thus, if the series diverges for every z ≠ ζ0 we put R = 0. On the other hand, if (1) is convergent for all values of z, we take R = ∞. If R > 0, we write
The radius of convergence may be determined by taking either lim |cn+1/cn| or lim as n → ∞ : if the limit is finite and equals L > 0 then R = 1/L, otherwise R = ∞. According to Taylor's theorem, every function f(z), which is differentiable in a domain D has a power series expansion of the form
which is unique for every ζ0 in D. This is called the Taylor series expansion of f(z) around the point ζ0 and its radius of convergence is the largest number R such that all z with |z − ζ0| < R lie within D. A Taylor series with center ζ0 = 0 is called a Maclaurin series.
Hille, E. (1973) Analytic Function Theory, Vol.1, Chelsea Publishing Co., New York.
Kreyszig, E. (1983) Advanced Engineering Mathematics, John Wiley, New York.