Infinite Sums, Infinite Products, and ζ(2k)

The most basic concept is that of an infinite sequence (of real or complex numbers in these notes). For p ∈ Z, let Np = {k ∈ Z : k ≥ p}. An infinite sequence of (complex) numbers is a function a : Np → C. Usually, for n ∈ Np we write a(n) = an, and denote the sequence by a = {an}n=p. The sequence {an}n=p is said to converge to the limit A ∈ C provided that for each > 0 there is an N ∈ Z such that |an − A| < for all n ∈ Z with n > N . When this holds we write limn→∞an = A. Def. A sequence {an}n=p of complex numbers is called a Cauchy sequence provided that for each > 0 there is some n0 ∈ Z such that for m,n ∈ Z,m ≥ n0 and n ≥ n0 imply that |am − an| < . When this is so, we write limm,n→∞|am − an| = 0.