1549901369-Elements_of_Real_Analysis__Denlinger_

8.7 Analytic Functions 531 (b) \In E .N, jtl Cj = jtl c~ aij) = i~ Ct 1 aij) (by the linearity of series, Theorem 8.1.12) ~ f (f ...

532 Chapter 8 • Infinite Series of Real Numbers We have just seen that all the row sums of both of these nonnegative matrices (d ...

7 Analytic Functions 533 00 00 To see that 2:: Ri and 2:: Cj converge absolutely, observe that 'Vn E .N, i=l j=l i~ IRil = i~ ...

534 Chapter 8 • Infinite Series of Real Numbers Prove the following slightly stronger version of Theorem 8.7.5: If f is infinit ...

8.8 *Elementary Transcendental Functions (Project) 53 5 00 For the double series I: aij defined by the matrix i,j=l 0 -1 0 0 0 ...

536 Chapter 8 • Infinite Series of Real Numbers Recall that a real number x is called an algebraic number if it is a solution of ...

8.8 *Elementary Transcendental Functions (Project) 537 functions and derive their familiar properties as if we had never encount ...

538 Chapter 8 • Infinite Series of Real Numbers (d) Vn EN, E(n) =en, and Vm E Z , E(m) =em. (e) Vn EN, E(l/n) = e^1 fn, and Vr E ...

8.8 *Elementary 'Iranscendental Functions (Project) 539 (d) 'Vx E JR., C(x + y) = C(x)C(y) - S(x)S(y); (e) 'Vx E JR., S(x - y) = ...

540 Chapter 8 n Infinite Series of Real Numbers Definition 8.8.16 7f = 2u, where u = min{t > 0: C(t) = O}. (That is, ~ is the ...

Chapter 9 Sequences and Series of Functions The chief concern of this chapter is uniform convergence and its consequences. By us ...

542 Chapter 9 • Sequences and Series of Functions On this set :F(S, JR) we define algebraic operations. For every pair of functi ...

9.1 Families of Functions and Pointwise Convergence 543 (9) VJ E F(S, JR), V r,s E JR, r(sJ) = (rs)J = s(rJ); (10) VJ E F(S, JR) ...

544 Chapter 9 • Sequences and Series of F\mc;tions Examples 9.1.5 The following are among the many subspaces of F(S, JR): (a) If ...

9 .1 Families of Tunctions and Pointwise Convergence 545 Examples 9.1. 7 We give here some examples of sequences {fn} and their ...

546 Chapter 9 11 Sequences and Series of Functions The graph of a typical f n is shown in Fig- ure 9.4. It is clear that the poi ...

9.1 Families of Functions and Pointwise Convergence 547 For each of the following sequences of functions Un}, find the largest ...

548 Chapter 9 • Sequences and Series of Functions sinnx \:/n EN and \:/x E JR, define fn(x) = fa. Prove that (a) {Jn} converge ...

9.2 Uniform Convergence 549 (See Figure 9.6.) Thus, l\fll is the distance between f and the 0 function. y y y=g(x) x x y=f(x) Fi ...

550 Chapter 9 • Sequences and Series of Functions Definition 9.2.3 (Uniform Convergence) We say that a sequence Un} of functions ...