An infinite series of the form

is called a "power series" expansion around the center ζ_{0} with constant "coefficients" c_{k}. The variable z and the constants ζ_{0} and ck may be real or complex numbers. For a given z = z_{0}, (1) becomes an infinite series of constant terms c_{n} (z_{0} − ζ_{0})^{n} which may or may not be convergent; in the first case, we denote the sum by S(z_{0}). We say that R is the "radius of convergence" of (1) if this series converges for all z with |z − ζ_{0}| < R and diverges for all z with |z − ζ_{0}| > R. Certainly, every series (1) converges at z = ζ_{0}. Thus, if the series diverges for every z ≠ ζ_{0} 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 |c_{n+1}/c_{n}| 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 ζ_{0} in D. This is called the *Taylor series* expansion of f(z) around the point ζ_{0} and its radius of convergence is the largest number R such that all z with |z − ζ_{0}| < R lie within D. A Taylor series with center ζ_{0} = 0 is called a *Maclaurin series.*

#### REFERENCES

Hille, E. (1973) *Analytic Function Theory,* Vol.1, Chelsea Publishing Co., New York.

Kreyszig, E. (1983) *Advanced Engineering Mathematics,* John Wiley, New York.