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.
References
- Hille, E. (1973) Analytic Function Theory, Vol.1, Chelsea Publishing Co., New York.
- Kreyszig, E. (1983) Advanced Engineering Mathematics, John Wiley, New York.