1549901369-Elements_of_Real_Analysis__Denlinger_

Answers & Hints for Selected Exercises 671 Case 2: x1 < xo. Revise the above argument to cover this case. (a) Suppose f ...

672 Appendix C • Answers & Hints for Selected Exercises Since the sequence { ~~ } is decreasing when n ::::: 2, I.Rn ( x) I ...

Answers & Hints for Selected Exercises 673 G'(x) = -g'(-x). Consequently, (a) F, G: - I -t IR where -I is an interval with l ...

674 Appendix C • Answers & Hints for Selected Exercises (i) For x > 0, f'(x) = ~e-lfx. Take p 1 (x) = x^2. Thus, the stat ...

Answers & Hints for Selected Exercises 675 EXERCISE SET 7.2 n n Since mi:::; Mi, S,_(f, P) = L mi:::; L Mi= S(f, P). i=l i= ...

676 Appendix C 11 Answers & Hints for Selected Exercises = l n-1 "' [ 9i^2 + Qi ] = l [ 9 n-1 "' i2 + 2. n-1 "' i ] = 27 (n- ...

Answers & Hints for Selected Exercises 6 77 Thus, s_u, Pn) = i~ l(xi)6.i = i~ ( 1 - (-1 + ~t) 3 ) · ~ = l ~ [2 _ ~ + 27i^2 _ ...

678 Appendix C • Answers & Hints for Selected Exercises EXERCISE SET 7 .3 Using the notation of Exercise 7.2.21, S..(f, Q) ...

Answers & Hints for Selected Exercises 679 (a) 1/ 2 (b) Let f(x) = x^2 • Then n k2 n 2 n 1 1 n-+lim oo iL =l 7i3 = n-lim Hx ...

680 Appendix C • Answers & Hints for Selected Exercises n n (a) For xi = Xi-1, R(f, Q~) = ~ I: l(xi_i) = ~ I: 1 = ; = 1. i=l ...

Answers & Hints for Selected Exercises 681 (a) f is bounded and continuous, hence integrable, on every [c, d] ~ (0, 1), so ...

682 Appendix C • Answers & Hints for Selected Exercises Let c > 0. Then 3 'P1 3 S..(f, 'P1) ?: f: f - ~ and 3 'P2 3 S_(g, ...

Answers & Hints for Selected Exercises 683 EXERCISE SET 7.6 If a= b, then J: f + J: = J: f + fca f = J: f - fca = 0 = J: f ...

684 Appendix C • Answers & Hints for Selected Exercises (b) [(x -t)k+^1 J(k+ll(t) - l J(k+l)(t)(k + l)(x -t)k(-dt)J: = [(x - ...

Answers & Hints for Selected Exercises 685 EXERCISE SET 7.8-A Suppose f integrable on every [c, b] such that a < c < ...

686 Appendix C • Answers & Hints for Selected Exercises ( ) e J" (n+1)"^1 sin x xi d x = L.; .;:.... Jk11: r(k+l)" I sin x x ...

Answers & Hints for Selected Exercises 687 discontinuous. By Lebesgue's criterion (7.9.5), the set of points of discontinuit ...

688 Appendix C 11 Answers & Hints for Selected Exercises (b) The contrapositive of (a). (c) 2:(-l)n diverges, but (1 - 1) + ...

Answers & Hints for Selected Exercises 689 (b) For p = 1 the series diverges (see Exercise 8.2.16). (c) For p < 1, the co ...

690 Appendix C • Answers & Hints for Selected Exercises L et ak = i.3.5.7 2·4·6 ·8 ....... .. (2k-1) ·(2k) · L et b k - l ...