A First Course in Probability and Markov Chains

Appendix A

Power series

We recall some results from the theory of power series which should already be known to the reader. Proofs and further results can be found in textbooks on Analysis, see e.g. [25, 26].

A.1   Basic properties

Let {a_n} be a sequence of complex numbers. The power series centred at zero with coefficients {a_n} is

i.e. the sequence of polynomials {s_n(z)}_n

Given a power series , the non-negative real number ρ defined by

where 1/0⁺ = +∞ and 1/(+∞) = 0, is called the radius of convergence of the series . In fact, one proves that the sequence {s_n(z)} is absolutely convergent if |z| < ρ and it does not converge if |z| > ρ. It can be shown that ρ > 0 if and only if the sequence {|a_n|} is not more than exponentially increasing. In this case, the sum of the series

is defined in the disc {|z| < ρ}.

Power series can be integrated and differentiated term by term. More precisely, ...