1549901369-Elements_of_Real_Analysis__Denlinger_

8.2 Nonnegative Series 471 If n = 2k + 1, 1 {l l/(2k+l) -k 1 1 1 (J3) 2k:+.l ~= 2k+yf,z = (3-k) = 32k+1 = 3 22k+1-2 = J3 1 -+- J ...

472 Chapter 8 • Infinite Series of Real Numbers Since the sum on the left is telescoping, this says Thus, the partial sums of L ...

8.2 Nonnegative Series 473 To find lim k (1 -ak+l) = lim k [1 -(k kP ) ] , we use L'Hopital's k-+oo ak k-+oo + 1 p rule: 1 imx. ...

(^474) Chapter 8 • Infinite Series of Real Numbers 8. v'2 v'4 v'6 JS JI5 3. 5 + 5. 7 + 7. 9 + 9. 11 + 11. 13 + ... 1 1·2 1 ·2· 3 ...

8.2 Nonnegative Series 475 00 1 Prove that ; n(ln n) [ln(ln n) JP converges if and only if p > l. L oo Inn Prove that - co ...

476 Chapter 8 • Infinite Series of Real Numbers ~1·3·5· .... (2k-1) Determine whether the series L.....,, 4 . 6 . 8 ..... ( 2 k ...

8.3 Series with Positive and Negative Terms 477 ALTERNATING SERIES Definition 8.3.1 If {an} is a sequence of positive numbers, t ...

478 Chapter 8 • Infinite Series of Real Numbers 00 Therefore, {Sn} converges (see Exercise 2.6.7). That is, .2,)-l)n+lan con- n= ...

8.3 Series with Positive and Negative Terms 479 and 0 < S2n+1 - S = azn+2 - azn+3 + azn+4 - azn+5 + azn+6 - · · · < azn+2· ...

480 Chapter 8 • Infinite Series of Real Numbers CAUTION: In using the alternating series test, students often ask whether it is ...

8.3 Series with Positive and Negative Terms 481 Theorem 8.3.10 Given any series I: an of real numbers, (a) I: an converges absol ...

482 Chapter 8 • Infinite Series of Real Numbers REARRANGEMENTS AND SUBSERIES We shall now explore some deeper consequences of ab ...

8.3 Series with Positive and Negative Terms 483 (c) Let I; an be an absolutely convergent series, and use the same notation used ...

484 Chapter 8 • Infinite Series of Real Numbers second negative term. Continue by induction. In general, after the nth negative ...

8.3 Series with Positive and Negative Terms 485 Theorem 8.3.16 A series converges absolutely if and only if each of its sub- se' ...

486 Chapter 8 n Infinite Series of Real Numbers Sum of the Alternating Harmonic Series: Let {Sn} denote the sequence of partial ...

8.4 The Cauchy Product of Series 487 terms, subtracting the next two terms, and so on. How does this differ from the series in E ...

488 Chapter 8 • Infinite Series of Real Numbers If we collect terms along the northeast-to-southwest diagonals, we have (a1 + a2 ...

8.4 The Cauchy Product of Series 489 Example 8.4.2 (Two Convergent Series Whose Cauchy Product Series Di- verges) Let I: ak and ...

490 Chapter 8 • Infinite Series of Real Numbers Note that (a) Each ck is obtained by grouping k successive terms of the series I ...