28 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS Exercise 1.10.24. (i) Prove the comparison test. (ii) If the series ∑ ∞ n=1 an converges to s and c is any constant, show that the series ∑ ∞ n=1 can converges to cs. (iii) Suppose that ∑ ∞ n=1 an and ∑ ∞ n=1 bn are infinite series. Suppose that an 0 and bn 0 for n ∈ N and limn→∞ an/bn = c 0. Show that ∞ n=1 an converges if and only if ∑ ∞ n=1 bn converges. The most useful series for comparison is the geometric series defined by a real number r, with 0 r 1. Theorem 1.10.25 (Ratio test). Suppose that ∑ ∞ n=1 an is a series of nonzero complex numbers. If r = limn→∞ |an+1/an| exists, then the series converges absolutely if r 1 and the series diverges if r 1. Proof. Suppose limn→∞ |an+1/an| = r 1. If ρ satisfies r ρ 1, then there exists N ∈ N such that |an+1|/|an| ρ for all n ≥ N. Consequently, |an| ≤ |aN|ρn−N for all n ≥ N. The result follows from the comparison test. Exercise 1.10.26. Show that if r 1, then the series above diverges, while, if r = 1, anything can happen. Our final test for convergence is called the root test. This can be quite effective when the comparison test and ratio test fail. Theorem 1.10.27 (Root test). Suppose that ∑ ∞ n=1 an is a series of complex numbers. Let r = lim sup n→∞ |an|1/n (consult Chapter 2 for a discussion of lim sup). If r 1, then the series converges absolutely. If r 1, then the series diverges. Proof. Suppose that lim sup n→∞ |an|1/n = r 1. Pick ρ so that r ρ 1. Then, there exists N ∈ N such that |an| ≤ ρn for all n ≥ N. The convergence of the series now follows from the comparison test. Exercise 1.10.28. Show that if r 1, then the above series diverges, while, if r = 1, anything can happen. Exercise 1.10.29. Suppose that the ratio test applies to a series. That is, limn→∞ |an+1|/|an| = r. Show that the lim sup n→∞ |an|1/n = r. Definition 1.10.30. Let z0 be a fixed complex number. A complex power series around z0 is a series of the form ∑ ∞ n=0 an(z − z0)n, where the coeﬃcients an ∈ C. When this series converges, it converges to a function of the complex variable z. Exercise 1.10.31. Show that if the series converges absolutely for a complex number z, then it also converges for any complex number w such that |w − z0| ≤ |z − z0|. That is, the series converges on the disk {w ∈ C | |w − z0| ≤ |z − z0|}.

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