1549901369-Elements_of_Real_Analysis__Denlinger_

Answers & Hints for Selected Exercises 691 (^16). Th esenes. 1 + 2 1 1 - 3 + 4 1 1 + 5 - 61 + 7 1 1 1 + 8 - 9 +··· d' iverge ...

692 Appendix C • Answers & Hints for Selected Exercises (b) "£bk converges by the alternating series test, but not absolutel ...

Answers & Hints for Selected Exercises 693 Assume t -1-2nn is fixed. By Lemma 8.5.4(b), £::cos kt= sin(n+ 21 ~~):;;sint/^2 ...

694 Appendix C • Answers & Hints for Selected Exercises 00 (a) L (-x)3k k=O on(-1,1) (c) ~ L., (-1)k k+l xk+3 k=O on (-1, ...

Answers & Hints for Selected Exercises 695 cos x = cos[c + (x - c)] =cos ccos(x - c) - sin csin(x - c) oo (-l)k(x-c)2k. oo ...

696 Appendix C • Answers & Hints for Selected Exercises Chapter 9 EXERCISE SET 9.1 (a) and (c), are subspaces; the others a ...

Answers & Hints for Selected Exercises 697 (a) \:In E N, [[fn[[ = +oo, so convergence is not uniform on R But on any compac ...

698 Appendix C • Answers & Hints for Selected Exercises ( <=) Suppose the hypotheses. Let c: > 0. Then 3no E N 3 m, n ...

Answers & Hints for Selected Exercises 699 On [O, 2] the limit function is f(x) = 0 if 0 < x::; 2, 1 if x = 0. Then 'in, ...

700 Appendix C • Answers & Hints for Selected Exercises (c) Since this is true Vx ~a> 0, it is true for all x > 0. App ...

Answers & Hints for Selected Exercises 701 p Q P=? Q (P =? Q) =? P) T T T T T F F T F T T F F F T F (not a tautology) p Q P ...

702 Appendix C • Answers & Hints for Selected Exercises "'"' (P V "'"' Q) := "'"'P A "'"'"'"' Q := "'"'PA Q "'"' [(P V Q) = ...

Answers & Hints for Selected Exercises 703 :Jx, 3y 3 [S(x,y) /\"' C(x,y)]; domain of x and y ={triangles} :Jx 3 Vy, [(y "I- ...

704 Appendix C • Answers & Hints for Selected Exercises Negation: :lx E I 3 Vy E 0 , x + y tJ. E. There is an integer such t ...

Answers & Hints for Selected Exercises 705 (a) n(-n,n)=(-1,1) LJ (-n,n) = (-oo,+oo) nEN nEN u (-n, n)c = (-oo, -1] u [1, +o ...

706 Appendix C • Answers & Hints for Selected Exercises (a)~ (b) (-oo,4] (c) [3,4] (d) [-2,-v'3]u[v'3,2] (e) (2, 4) (i) (-2 ...

Answers & Hints for Selected Exercises 707 Let J, g E F(S, ~).Then, Vx ES, (! + g)(x) = J(x) + g(x) = g(x) + J(x) = (g + f) ...

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Bibliography [1] Abbott, Stephen, Understanding Analysis. New York: Springer-Verlag, 2001. [2] Al-Gwaiz, M. A., and S. A. Elsano ...

710 Bibliography [13] Bilodeau , Gerald G., G. E. Keogh, and Paul R. Thie, An Introduction to Analysis, Second Edition. Sudbury, ...