26 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS a sum of an infinite number of elements from some place or other. These elements could be numbers, functions, or what have you, so we begin with one-sided series of numbers. An infinite series is an expression of the form ∑ ∞ n=1 an, where the ele- ments an come from a number system in which addition makes sense. So that we don’t wander around aimlessly, let’s fix our number system to be the complex numbers, that is, an ∈ C, with the possibility of restricting our- selves to the real numbers or even the rational numbers. In the definition we have chosen to use the natural numbers as the index set, but in consider- ing infinite series, we could start the summation with any integer and write ∞ n=n0 an. Later, we will also consider two-sided series where the index set is the integers and we write ∑ ∞ −∞ an. If these expressions are going to have any meaning at all, we must look at the partial sums. Definition 1.10.11. If ∑ ∞ n=1 an is an infinite series of complex numbers, the N-th partial sum of the series is SN = ∑ N n=1 an. Examples 1.10.12. (i) Let an = 1 for all n. Then SN = N. (ii) Let an = 1/n. Then SN = 1 + 1/2 + · · · + 1/N. (iii) Let an = 1/2n. Then SN = 1 − 1/2N. (iv) Let an = (−1)n+1. In this case, SN = 1 if N is odd and 0 if N is even. (v) Fix θ, with 0 θ 2π, and let an = einθ/n. Then SN = ∑ N n=1 einθ/n, which is the best we can do without more information about θ. (vi) Let an = sin nπ/n2. In this case, SN = ∑ N n=1 sin(nπ)/n2. Definition 1.10.13. Let ∑ ∞ n=1 an be an infinite series of complex num- bers. The sequence (SN)N∈N is called the sequence of partial sums. We say that the series ∑ ∞ n=1 an converges if the sequence of partial sums (SN)N∈N converges. If the sequence (SN)N∈N does not converge, we say that ∞ n=1 an diverges. Of course, since we are working in C, the series converges if and only if the sequence (SN)N∈N is a Cauchy sequence. That is, given ε 0, there is an N ∈ N so that for n, m N (assuming n m), | n k=m+1 an| ε. Exercise 1.10.14. Determine which of the series in Example 1.10.12 converge. We are faced with two problems. The first is, “How do we tell if a series converges?” The second is, “If a series does converge, how do we find the explicit sum?” There is extensive literature about these two questions, but the fact is that the second question presents many more diﬃculties than the first. In Chapter 7, the theory of Fourier series will provide some assistance. The most helpful series in all of this discussion is the geometric series. Definition 1.10.15. Let z be a complex number. The geometric series defined by z is ∑ ∞ n=0 zn.

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