1549901369-Elements_of_Real_Analysis__Denlinger_

8.4 The Cauchy Product of Series 491 is a partial sum of a rearrangement of L dk. By Theorem 8.3.13, every rear- rangement of an ...

492 Chapter 8 11 Infinite Series of Real Numbers Then, n ~no=} ltakBn+l-kl < t lan+I-kBkl k=l k=l no n = L lan+i-kBkl + L lan ...

8.4 The Cauchy Product of Series 493 series converges when lrl < 1, and find its sum. Verify the conclusion of Theorem 8.4.3 ...

494 Chapter 8 • Infinite Series of Real Numbers By Theorem 8.4.4, L ck must converge. Prove that this convergence is not absolut ...

8.5 Series of Products 495 Thus, \:/ 1 ::; m < n, n n n L akbk = L Sk(bk - bk+1) + L (Skbk+I - Sk-1h) k=m k=m k=m n L Sk(bk - ...

496 Chapter 8 • Infinite Series of Real Numbers Therefore, by the Cauchy criterion for convergence of series (Theorem n 8.1.11), ...

8.5 Series of Products 497 n Lemma 8.5.5 (a) Vt E JR, the partial sums I:: sin kt are bounded. k=l n (b) Vt f=. 2p7r, (p E Z), t ...

498 Chapter 8 • Infinite Series of Real Numbers Theorem 8.5.8 (Abel's Test) If L;ak converges and {bk} is a bounded, monotone se ...

8.5 Series of Products 499 Vx\ '[/,TE !Rn, r + ('[/ + 7) = (r + '[/) + 7. Vr, '[/ E !Rn, r + '[/ = '[/ + r. 3 (}' E !Rn 3 Vr E ...

500 Chapter 8 • Infinite Series of Real Numbers linear algebra course, we omit them here. Because of property 4, given any J! E ...

8.5 Series of Products 501 SQUARE SUMMABLE SEQUENCES We are seeking conditions on the series I.: ak and I.: bk, weaker than abso ...

502 Chapter 8 • Infinite Series of Real Numbers n EN. Define a'= (la1l,la2I,··· ,lanl) and f! = (lbil,lb 2 I,··· ,lbnl). Then , ...

8.5 Series of Products 503 m = 1, and using the sequence {xk} instead of {ak}, show that the n n conclusion of Theorem 8.5.2 bec ...

504 Chapter 8 • Infinite Series of Real Numbers Prove that if { ak} is square summable, then lim ak = 0. k->oo Prove that if ...

8.6 Power Series 505 it is always an interval centered at c. The next theorem is the first step in understanding this situation. ...

506 Chapter 8 • Infinite Series of Real Numbers Corollary 8.6.4 The set of points x for which a given power series 00 L ak(x - c ...

8.6 P ower Series 507 the radius of convergence is 0, while if the series converges for all real numbers, we say that the radius ...

508 Chapter 8 • Infinite Series of Real Numbers The root test (8.2.15) may also be used in finding the interval of convergence o ...

8.6 Power Series 509 00 Definition 8.6.10 Given a power series L ak(x - c)k, the power series ob- k=O 00 tained from it by diffe ...

510 Chapter 8 • Infinite Series of Real Numbers Proof. Exercise 6. 0 In the next group of results we show that a function repres ...