1549901369-Elements_of_Real_Analysis__Denlinger_

9.2 Uniform Convergence 551 (c) In Example 9.1.7 (c), the sequence {fn} of functions { l. if lxl < l. · fn(x) = l;I if ~ < ...

552 Chapter 9 • Sequences and Series of Functions using the squeeze theorem. In summary, 3 no E .N 3 m, n 2 no=> \fx ES, lfn( ...

9.2 Uniform Convergence 553 00 Theorem 9.2.11 (Uniform Cauchy Criterion for Series) A series I: fk k=O of functions in F(S, JR) ...

554 Chapter 9 • Sequences and Series of Functions 00 Corollary 9.2.15 A power series I: ak(x - c)k converges uniformly (and ab- ...

9.2 Uniform Convergence 555 llgkll = fc ---+ 0, so gk ---+ 0 uniformly on [o, ?T - o]. Therefore, by Dirichlet's -^00 sinkx test ...

556 Chapter 9 • Sequences and Series of Functions Determine whether the following sequences {1 n} converge uniformly on [O, +oo ...

9.3 Impli cations of Uniform Convergence in Calculus 557 9.3 Implications of Uniform Convergence in Calculus It is natural to in ...

558 Chapter 9 • Sequences and Series of Functions (b) Let S = [O, l] and fn(x) = xn. As we saw in Example 9.1.7 (a), {fn} conver ...

9.3 Implications of Uniform Convergence in Calculus 559 The graph of a typical f n is shown in Fig- ure 9.11. The pointwise limi ...

560 Chapter 9 • Sequences and Series of Functions Since this is true Vm, n ;::: n 0 , {Ln} is a Cauchy sequence, and so it must ...

9.3 Implications of Uniform Convergence in Calculus 561 Theorem 9.3.8 (Uniform Convergence Preserves Integrals) If fn--> f un ...

562 Chapter 9 • Sequences and Series of Functions Solution: Let 0 < a < n. On the interval [a, n], II si~;x II < ~a -t ...

9.3 Implications of Uniform Convergence in Calculus 563 Then, lfn(x) - fm(x)I :=::; lfn(xo) - fm(xo)I +Ix -xol lf~(t) -f,',,(t)I ...

564 Chapter 9 • Sequences and Series of Functions Let x =f x 0 in [a,b]. By Equation (4), 3t between x and xo such that I [fn(x) ...

9.3 Implications of Uniform Convergence in Calculus 565 00 Example 9.3.13 L f< sin~ converges uniformly on any compact interv ...

566 Chapter 9 • Sequences and Series of Functions Fix an no EN. 'Vx ES, Un(x)} is monotone decreasing, so whenever nk ~no, f no ...

9.3 Implications of Uniform Convergence in Calculus 567 Evaluate each of the following integrals and justify your answers: ( ) ...

568 Chapter 9 • Sequences and Series of Functions 00 1 Riemann's Zeta function^8 is defined ((x) = I:; kx, for all x > 1. k= ...

9.4 *Two Results of Weierstrass 569 AN EVERYWHERE CONTINUOUS, NOWHERE DIFFERENTIABLE FUNCTION In 1872 , Weierstrass presented a ...

570 Chapter 9 • Sequences and Series of Functions (14) and subsequently constructed his own example, the function (12) above. In ...