1549901369-Elements_of_Real_Analysis__Denlinger_

8.6 Power Series 511 Writing the sum within braces in reverse order , this sum becomes 00 L ak{xk-2(y-x) + xk-3(y2 x2) + xk-4(y ...

512 Chapter 8 • Infinite Series of Real Numbers (b) f has derivatives of all orders at every x in the interior of I. 00 (c) For ...

8.6 Power Series 513 Examples 8.6.18 Maclaurin Series for ln(l + x), tan-^1 x, ( 1 ) 2 and ( x 2. l+x l+x) From our knowledge of ...

514 Chapter 8 • Infinite Series of Real Numbers ( Thus, tan-^1 x = x - x^3 5 7 ) 3 + x^5 - x^7 + · · · +C. Letting x = 0, we fin ...

8.6 Power Series 515 00 Theorem 8.6.19 (Abel's Theorem) Suppose f(x) lx l < l. L akxk for all k=O 00 00 (a) If L ak converges ...

516 Chapter 8 • Infinite Series of Real Numbers Choose 6 < min { 1, 2 ~ }. Then, l-6<x<l=?l-x<6 € € € =? (1-x)M + 2 ...

8.6 Power Series 517 (b) Applying similar reasoning to Example 8.6.18 (b), we can show that the Maclaurin series for tan-^1 x is ...

518 Chapter 8 • Infinite Series of Real Numbers Complete Part 2 of the proof of Theorem 8.6.14. Find the Taylor series for ln x ...

8.7 Analytic Functions 519 Use Abel's theorem to prove Abel's claim that if the Cauchy product of two convergent series converg ...

520 Chapter 8 • Infinite Series of Real Numbers c and x, where x -:/:- c, and f(n+l)(t) exists for all t in the open interval I ...

8.7 Analytic Functions 521 Proof. (a) See Examples 6.5.5 and 6.5.13. (b) One can easily verify that the Maclaurin polynomials ar ...

522 Chapter 8 • Infinite Series of Real Numbers Using the Maclaurin series for ex and the algebra of power series (8.6.9), we fi ...

8.7 Analytic Functions 523 where (~) = 1 and (~) = a(a - l)(a -^2 1l· .. (a - k + l) when k ;::: 1, is called the binomial serie ...

524 Chapter 8 • Infinite Series of Real Numbers Then, Vk E N, the kth Maclaurin coefficient is J(k) (0) = (a). k! k 00 Therefore ...

7 Analytic Functions 525 Case 2 (a< 1): Let M = max{l, ll + x l°'-^1 }. Arguing as in Case 1 above (Exercise 10) we can sho ...

526 Chapter 8 a Infinite Series of Real Numbers J dx Proof. Recall that sin-^1 x = .Jl=X2' From Example 8.7. 11 , we have 1-x^2 ...

8.7 Analytic Functions 527 function f to be analytic at cits Taylor series must converge to f(x) in some neighborhood of c. Thus ...

528 Chapter 8 • Infinite Series of Real Numbers 00 f(x) = L ak[(d - c) + (x - d)]k and, using the binomial theorem, k=O = L^00 a ...

7 Analytic Functions 529 k (k = 0, 1, 2, 3,-· · ) is [j~O e;j)ak+j(d-c)jl (x-d)k. (We know that t he column sums exist because ...

530 Chapter 8 • Infinite Series of Real Numbers 00 is called the sum by rows of L aij. If all the column sums (series) Cj i ,j=l ...