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Function Sequences and Series Chapter 7 Function Sequences and Series 7.1 Complex Numbers This is not meant to be a thourogh and completely rigorous construction of complex numbers but rather a quick brush up of things assumed to be already known. 7.1.1 Basic Properties The field of complex number can be viewed as C = R + iR but is also con- 2 venient to have a geometric model like C = R . The next table summarizes the basic field operations in the two different model 2 C = R + iR C = R z = x + iy z = (x, y) x = <(z) , y = =(z) x = <(z) , y = =(z) z1 + z2 = (x1 + x2) + i(y1 + y2) z1 + z2 = (x1 + x2, y1 + y2) z1z2 = x1x2 − y1y2 + i(x1y2 + x2y1) z1z2 = (x1x2 − y1y2, x1y2 + x2y1) i2 = −1 (0, 1)(0, 1) = (−1, 0) (C , + , ·) is an algebraically closed field. We also recall the simple formula 1 x y = − i , x 6= 0 6= y . x + iy x2 + y2 x2 + y2 Further we define the absolute value (or modulus) of a complex number z by p |z| := x2 + y2 93 94 CHAPTER 7. FUNCTION SEQUENCES AND SERIES and mention the simple properties |z1 z2| = |z1| |z2| , |z1 + z2| ≤ |z1| + |z2| , z1 , z2 ∈ C . What are their geometric meaning? Definitions 7.1.1. (Convergence) Let (zn)n∈N be a sequence in C. (i) We say that it converges in C towards a limit z∞ ∈ C iff zn → z∞ (n → ∞) ∀ ε > 0 ∃ N ∈ N s.t. |zn − z∞| ≤ ε ∀ n ≥ N. (ii) We call it a Cauchy sequence iff ∀ ε > 0 ∃ N ∈ N s.t. |zn − zm| ≤ ε ∀ m, n ≥ N. (iii) A function f : R → C is called complex-valued. In this case f = g + ih for g, h : R → R . o Definition 7.1.2. A complex-valued function f : Df ⊂ R → C is said to 0 be differentiable at x0 ∈ Df iff there exists f (x0) ∈ C such that 0 f(x0 + h) = f(x0) + f (x0)h + o(|h|) . It is simply called differentiable if it differentiable everywhere and continu- 0 ously differentiable if its derivative f : Df → C is continuous. Continuity of a complex-valued function is defined similarly to that of real-valued function, where the absolute valued has to be substituted by the modulus. Finally f is R-integrable if any of the conditions of theorem 6.7.6 satisfied, where convergence is now in C. Remarks 7.1.3. (a) Observe that ( xn → x∞ zn → z∞ (n → ∞) ⇐⇒ (n → ∞) yn → y∞ for z∞ = x∞ + iy∞. (b) Using the fact that f = g + ih, many properties of complex-valued functions can be reconducted to properties of g and h. In particular f is continuous at x0 ∈ Df ⇐⇒ g, h are continuous at x0 . (7.1.1) f is differentiable at x0 ∈ Df ⇐⇒ g, h are differentiable at x0 . (7.1.2) m Tm(f, x0) = Tm(g, x0) + iTm(h, x0) , f ∈ C Df , C) . (7.1.3) f ∈ R(a, b; C) ⇐⇒ g, h ∈ R(a, b) . (7.1.4) 7.2. NUMERICAL SERIES AND SEQUENCES 95 Theorem 7.1.4. Let f ∈ R(a, b : C), then |f| ∈ R(a, b) and Z b Z b | f(x) dx ≤ |f(x)| dx . a a Proof. The claim follows by the inequality |z1| − |z2| ≤ |x1 − x2| + |y1 − y2| , which follows from the triangular inequality and which implies that Osc(|F |, P) ≤ osc(f, P) + osc(g, P) and |S(F, P)| ≤ S(|F |, P) . √ 7.2 Numerical series and Sequences 7.2.1 Convergence and Absolute Convergence P∞ We begin by remarking that the convergence of a series k=1 xk can simply be viewed as the convergence of the sequence (sn)n∈N of its partial sums Pn sn = k=1 xk. In this sense a series is nothing but a sequence itself. P∞ Definition 7.2.1. A series k=1 xk of complex numbers xk ∈ C , k ∈ N is P∞ called absolutely convergent iff k=1 |xk| is convergent. Example 7.2.2. (Geometric Series) P∞ k Let r ∈ [0, ∞) and consider the geometric series k=1 r . Considering its partial sums, we easily obtain that n X 1 − rn+1 rk = 1 − r k=1 and conclude that convergence is given if r < 1. What is the limit (or value of the series)? Proposition 7.2.3. (Elementary Properties) P∞ P∞ Consider two convergent series k=1 xk and k=1 yk of complex numbers. Then, for λ ∈ C, ∞ ∞ ∞ ∞ X X X X (xk + yk) converges and (xk + yk) = xk + yk , (7.2.1) k=1 k=1 k=1 k=1 ∞ ∞ ∞ X X X λxk converges and λxk = λ xk . (7.2.2) k=1 k=1 k=1 96 CHAPTER 7. FUNCTION SEQUENCES AND SERIES P∞ Theorem 7.2.4. k=1 xk converges iff n X ∀ ε > 0 ∃ N(ε) such that xk ≤ ε ∀ m, n ≥ N(ε) . k=m Proof. Since is complete and the condition above implies that the sequence C √ of partial sums is a Cauchy sequence, its converence gives the claim. P∞ Remarks 7.2.5. (a) If k=1 xk converges, then xk → 0 (k → ∞). (b) Absolute convergence implies convergence since n n X X xk ≤ |xk| ∀m, n ∈ N . k=m k=m Theorem 7.2.6. (Comparison Test) P∞ P∞ Let k=1 xk and k=1 yk be given such that |xk| ≤ yk , k ∈ N . P∞ P∞ Then, if k=1 yk converges, the series k=1 xk is absolutely convergent. Proof. The inequality n n n X X X xk ≤ |xk| ≤ yk ∀ m ≤ n ∈ N k=m k=m k=m implies that P∞ |x | satisfies Cauchy criterion if P∞ y does. If P∞ y k=1 k √ k=1 k k=1 k converges, that is clearly the case. Corollary 7.2.7. (Ratio and Root Tests) xk+1 (i) ∃ N ∈ N , r ∈ (0, 1) such that | | ≤ r < 1 ∀ k ≥ N xk ∞ X =⇒ xk converges absolutely. k=1 The series diverges, if N ∈ can be found such that | xk+1 | ≥ 1 for all N xk k ≥ N. 1/k (ii) ∃ N ∈ N , r ∈ (0, 1) such that |xk| ≤ r < 1 ∞ X =⇒ xk converges absolutely. k=1 7.2. NUMERICAL SERIES AND SEQUENCES 97 P∞ 1 Example 7.2.8. The series k=1 ka converges for a > 1 and diverges for a ≤ 1. Definition 7.2.9. (Rearrangement) Let π : N → N be a permutation, then ∞ X xπ(k) k=1 P∞ is called rearrangement of k=1 xk. It is natural to ask whether a convergent sequence preserves its conver- gence after a rearrangement. The next two theorems answer precisely this question. P∞ Theorem 7.2.10. (i) If k=1 xk is absolutely convergent and π is any P∞ permutation of the naturals, then k=1 xπ(k) converges absolutely as well and has the same value. P∞ (ii) If k=1 xπ(k) converges regardless of the choice of permutation π, then P∞ k=1 xk is absolutely convergent. P∞ Proof. Let π be a rearrangement. Since k=1 xk is absolutely convergent, for any ε > 0 we can find N˜ ∈ N such that ∞ X |xk| ≤ ε . k=N˜+1 Then choose N ∈ N such that {x1, . , xN˜ } ⊂ {xπ(1), . , xπ(N)} . In that case n N˜ ∞ X X X xπ(k) − xk ≤ |xk| ≤ ε , ∀ n ≥ N k=1 k=1 k=N˜+1 and consequently ∞ n ∞ N˜ N˜ n X X X X X X xk − xπ(k) ≤ xk − xn + xn − xπ(k) k=1 k=1 k=1 k=1 k=1 k=1 ≤ 2ε , ∀ n ≥ N 98 CHAPTER 7. FUNCTION SEQUENCES AND SERIES which gives the desired convergence. P∞ (ii) We prove the converse. Assume that k=1 |xk| is divergent. The we P∞ need to find at least a permutation π such that k=1 xπ(k) diverges, too. Without loss of generality we can assume that ∞ X xk ∈ R ∀ k ∈ N and ∃ (kn)n∈N with xkn > 0 and xkn = ∞ . n=1 To get a proper rearragement we still need to include the terms which are ˜ still missing. Let (kn)n∈N be the sequence of the indeces missed by (kn)n∈N. Notice that it might be finite. Then define N1 such that N X1 s + x := x + x ≥ 1 1 k˜1 kn k˜1 n=1 and squeeze x into (x ) right after x . Then proceed inductively k˜1 kn n∈N kN1 to choose Nm ∈ N for m ≥ 2 such that m−1 Nm X X s + x˜ + x + x˜ ≥ m l kl kn km l=1 n=Nm−1 and squeeze x˜ right after xk . In this fashion we obtain a rearrangement km Nm √ of the original series which diverges as well. P∞ Theorem 7.2.11. Let k=1 xk be a conditionally convergent series and x∞ ∈ R. Then there exists a permutation π of the naturals such that ∞ X xπ(k) = x∞ . k=1 Thus a conditionally convergent series can be made to converge to any limit by rearranging it. + − Proof. (i) Consider the two subsequences (xk )k∈N and (xk )k∈N comprising the positive and negative terms of (xk)k∈N, respectively. Then necessarily ∞ ∞ X + X − either xk = ∞ or xk = −∞ k=1 k=1 since, otherwise, the series would converge absolutely. ± (ii) Next we put together a new series recusively as follows. First find N1 7.2. NUMERICAL SERIES AND SEQUENCES 99 such that + + PN + N := inf N s := x > x∞ , 1 1 k=1 k N − + X − N1 := inf N s1 + xk < x∞ , k=1 | {z } − =:s1 and then, for m ≥ 2, Nm + − X + N := inf N s + x > x∞ , m m−1 k k=N + +1 m−1 | {z } + sm − Nm − + X − Nm := inf N sm + x < x∞ , k k=N − m−1 | {z } − sm (iii) Finally we observe that − + sm → x∞ and sm → x∞ (m → ∞) follows from + − |x − s | ≤ |x + | and |x − s | ≤ |x − | ∀m ∈ ∞ m Nm ∞ m Nm N √ combined with remark 7.2.5(a).
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