1549901369-Elements_of_Real_Analysis__Denlinger_

7.9 *Lebesgue's Criterion for Riemann Integrability 451 Hence, \:/c: > 0, Xe can b e squeezed between two step functions, 0 a ...

452 Chapter 7 • The Riemann Integral An example^23 of a bounded open set A such that XA is not integrable on any closed interva ...

Chapter 8 Infinite Series of Real Numbers Sections 8.1 and 8.2 cover basic concepts and the standard convergence tests. Section ...

454 Chapter 8 • Infinite Series of Real Numbers 00 00 If L an = +oo (or -oo) we do not say that I.: an converges, but rather say ...

8.1 Basic Concepts and Examples 455 a =f 0, the geometric series converges iff lrl < 1, and for all such r, oo a .Z::::: arn ...

456 Chapter 8 • Infinite Series of Real Numbers (^00 1 1 1 1) Example 8.1.6 The harmonic series '""' - = 1 + -+ -+ · · · + - + · ...

8.1 Basic Concepts and Examples 457 Theorem 8.1.10 (Grouping by Inserting Parentheses) (a) If'L, an converges, then any grouping ...

458 Chapter 8 • Infinite Series of Real Numbers EXERCISE SET 8.1 Prove that ~ L.., ---2_ 10n converges, and find its limit. Wha ...

8.2 Nonnegative Series 459 00 1 Prove that L 2 5 6 converges, and find its sum. n=l n + n + Prove that if x is not a negative ...

460 Chapter 8 • Infinite Series of Real Numbers 00 function such that lim f(x) = 0. Then the series L an converges ¢==:::} the X ...

8.2 Nonnegative Series 461 Thus, for all integers n > no, 1:: 0 f :S CEo ak) - an. (5) With the help of Figure 8.1 (b) we see ...

462 Chapter 8 m Infinite Series of Real Numbers 00 00 Thus, by the integral test, '""" n diverges. Therefore, '""" n di- n=4 ~ n ...

8.2 Nonnegative Series 463 COMPARISON TESTS Theorem 8.2.6 (Comparison Test) Suppose Lan and L bn are nonneg- ative series, and : ...

464 Chapter 8 • Infinite Series of Real Numbers Theorem 8.2.8 (Limit Comparison Test) Suppose Lan and L bn are nonnegative serie ...

8.2 Nonnegative Series 465 ~n. Then lim an = lim ( y'5rL - 10. yn) = lim ( v'5 n - lOyn) = y ' " n->oo bn n--+oo 3n + yn 1 n- ...

466 Chapter 8 • Infinite Series of Real Numbers 00 00 Apply the comparison test to the series L an and L an 0 rn. Since the n=no ...

8.2 Nonnegative Series 467 Solution. an+l. (n+l)^2 +1 2n. n^2 +2n+2 2n (a) lim = hm · --= hm · -- n-+oo an n-+oo 2n+l n2 + 1 n-+ ...

468 Chapter 8 • Infinite Series of Real Numbers THE ROOT TEST Theorem 8.2.14 (Root Test, Basic Form) Let I: an be a nonnegative ...

8.2 Nonnegative Series 469 Examples 8.2.17 Use the root test to test the following series for convergence or divergence: (a) f n ...

470 Chapter 8 • Infinite Series of Real Numbers Proof. Let {an} be a sequence of positive real numbers. an+l Part 1: We prove ...