POWER SERIES

An infinite series of the form

(1)

is called a "power series" expansion around the center ζ₀ with constant "coefficients" c_k. 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 c_n (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 |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 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.