# 03/17/2019: Advanced Calculus (Patrick Fitzpatrick) /home/zack/Dropbox/Library/Patrick Fitzpatrick/Advanced Calculus (613)/Advanced Calculus - Patrick Fitzpatrick.pdf Last Annotation: 03/17/2019 ## Notes - Full expression of Taylor Expansion (Patrick Fitzpatrick 219) - The remainder theorem for Taylor polynomials (Patrick Fitzpatrick 222) - Integral expression for the Taylor remainder (Patrick Fitzpatrick 235) - Extended Binomial formula (Patrick Fitzpatrick 236) - We can make the denominator as small as we want (Patrick Fitzpatrick 265) - Essentially the last theorem, just replacing with Cauchy condition (Patrick Fitzpatrick 271)
## Highlights - Definition Let I be a neighborhood of the point x 0 • Two functions f: I --+ lR and g: I --+ lR are said to have contact of order 0 at x 0 provided that f\(x 0 \) = g\(x0 \)\. (Patrick Fitzpatrick 218) - For a natural number n, the functions f and g are said to ha (Patrick Fitzpatrick 218) - ve contact of order n at x 0 provided that f : I --+ lR and g : I --+ lR have n derivatives and for 0::::: k::::: n\. (Patrick Fitzpatrick 218) - What is really surprising is that frequently it happens that lim [f\(x\)Pn\(x\)] = 0, n-+oo even when the point x is far away from x 0 • As we will show in Section 8\.6, it can also happen that the Taylor polynomials for certain functions do not provide good approx1 imations at any point x other than x 0 , no matter how large the index n (Patrick Fitzpatrick 221) - The sequence {\( -1 \)n} shows that, in general, it is not true that any bounded sequence converges (Patrick Fitzpatrick 247) - Definition A sequence of numbers {an} is said to be a Cauchy sequence (Patrick Fitzpatrick 247) - We will prove that a sequence of numbers converges if and only if it is a Cauchy sequence\. (Patrick Fitzpatrick 247) - Proposition 9\.5 Suppose that the series 2:::: an converges\. Then lim an = 0\. (Patrick Fitzpatrick 250) - As we have already seen in Chapter 2, the Harmonic Series 00 1 00 1 I:-n n n=1 does not converge despite the fact that limn~oo 1/n = 0\. (Patrick Fitzpatrick 250) - We have two principal general criteria for a sequence of numbers to converge, namely, the Monotone Convergence Theorem and the Cauchy Convergence Criterion\. (Patrick Fitzpatrick 251) - Theorem 9\.7 Suppose that {ak} is a sequence of nonnegative numbers\. Then the series 2:::~ 1 ak converges if and only if the sequence of partial sums is bounded; that is, there is a positive number M such that for every index n\. (Patrick Fitzpatrick 251) - The Monotone Convergence Theorem asserts that the sequence of partial sums converges if and only if the sequence of partial sums is bounded\. (Patrick Fitzpatrick 251) - Corollary 9\.8 The Comparison Test Suppose that {ad and {bd are sequences of numbers such that for index k, i\. The series 2:: 1 ak converges if the series 2:: bk converges\. ii\. The series 2:: bk diverges if the series 2:: ak diverges (Patrick Fitzpatrick 252) - Then the series 2:~ 1 ak is convergent if and only if the sequence of integrals {j n f \(x\) dx} is bounded\. (Patrick Fitzpatrick 252) - f : [ 1, oo\) ~ IR is continuous and monotonically (Patrick Fitzpatrick 252) - Since the function f is continuous, its restriction to each bounded interval is integrable (Patrick Fitzpatrick 253) - inequalities imply that the sequence of partial sums for the series 2:~ 1 ak is bounded if and only if the sequence {ft f \(x\) dx} is bounded\. (Patrick Fitzpatrick 253) - Therefore, in view of Theorem 9\. 7, it follows that the series 2:~ 1 ak is convergent if and only if the sequence {j1n f \(x\) dx} is bounded\. (Patrick Fitzpatrick 253) - Corollary 9\.13 The pTest For a positive numb er p, the series 00 1 LkP converges if and only if p > 1 (Patrick Fitzpatrick 254) - When the terms of a series fail to be of orie sign, it is not possible to directly invoke the Monotone Convergence Theorem (Patrick Fitzpatrick 254) - ically deTheorem 9\.15 The Alternating Series Test Suppose that {ak} is a monoton creasing sequenc e of nonnega tive numbers that converges to 0\. Then the series converges\. (Patrick Fitzpatrick 255) - By the Monoton e Converg ence Theorem , the sequence {s2n} converges (Patrick Fitzpatrick 255) - show first that the subseque nce {s2n} converge s\. (Patrick Fitzpatrick 255) - For series whose terms are neither of one sign nor alternating in sign, it is natural to apply the Cauchy Convergence Criterion for Sequences to the sequence of partial sums (Patrick Fitzpatrick 256) - It is sometimes useful, particularly when considering series, to restate the definition of a Cauchy sequence as follows: A sequence {sn} is a Cauchy sequence provided that for each positive number E there is an index N such that for each index n ::=::: N and any natural number k, (Patrick Fitzpatrick 256) - Theorem 9\.17 The Cauchy Convergence Criterion for Series The series 2:~ 1 ak converges if and only if for each positive number E there is an index N such that for all indices n ::=::: N and all natural numbers k\. (Patrick Fitzpatrick 256) - Corollary 9\.18 The Absolute Convergence Test An absolutely convergent series converges; that is, the series 2:~ 1 ak converges if the series 2:~ 1 lak I converges\. (Patrick Fitzpatrick 256) - Theorem 9\.20 For the series \.Z::::~ 1 ab suppose that there is a number r with 0 =::: r < 1 and an index N such that for all indices n ~ N\. \(9\.7\) Then the series \.Z::::~ 1 ak is absolutely convergent\. (Patrick Fitzpatrick 257) - Corollary 9\.21 The Ratio Test for Series For the series 2:::~ 1 ab suppose that lim ian+! I = f\.\. n--+oo ian I i\. If,£ < 1, the series 2:::: 1 an converges absolutely\. ii\. If,£ > 1, the series 2:::: 1 an diverges\. (Patrick Fitzpatrick 258) - Definition Given a function f: D -+ IR and a sequence of functions {fn: D -+ JR}, we say that the sequence {fn: D-+ IR} converges pointwise to f: D -+ IR, or that {fn} converges pointwise on D to f, provided that for each point x in D, lim fn\(x\) = f\(x\)\. (Patrick Fitzpatrick 260) - Observe that this is an example of a sequence of continuous functions that converges pointwise to a discontinuous function\. (Patrick Fitzpatrick 260) - Observe that this is an example of a sequence of functions, each of which is differentiable on IR, that converges pointwise on IR to a function that is not differentiable atx =0\. (Patrick Fitzpatrick 261) - This is an example of a sequence of integrable functions that converges pointwis e on a closed bounded interval to a function that is not integrable \(Exercise 5\)\. (Patrick Fitzpatrick 262) - Thus, the sequence of functions {fn} converges pointwise on the interval [0, 1] to 0 \(by this we mean to the function that is identically equal to 0 on [0, 1]\)\. Observe 1 \(by th J that 0 0 1 f = 0, while for each index n, J 0 fn = 1\. (Patrick Fitzpatrick 262) - Question A Suppose that each function fn : D --+ lR is continuous\. Is the limit function f : D --+ lR also continuous ? Answer: No\. Example 9\.22 describes a sequence of polynomials that converges pointwise on the interval [0, 1] to a discontinuous function\. (Patrick Fitzpatrick 264) - Question B If D = I is an open interval and each function fn : I --+ lR is differentiable, is the limit function f : I --+ lR also differentiable? If it is, is nction f : I --+ lR also diff \. [dfn hm l -\(x\) = -\(x\)? df n-+00 dx dx Answer: No\. Example 9\.23 describes a sequence of exponential functions that converges pointwise on lR to a nondifferentiable function\. (Patrick Fitzpatrick 264) - Question C If D =[a, b] and eachfuncti on fn: [a, b]--+ lR is integrable, is the limit function f: [a, b]--+ lR also integrable? !fit is, is }~ [ [ fn] = [ f? Answer: No\. Example 9\.24 describes a sequence of step functions that converges pointwise on the interval [0, 1] to a nonintegrable function\. Moreover, as Example 9\.25 shows, even if the limit function is integrable, it is not necessarily the case that the limit of the integrals equals the integral of the limit\. (Patrick Fitzpatrick 264) - Definition Given a function f: D --+ lR and a sequence of functions {fn: D --+ lR}, the sequence {fn: D --+ lR} is said to converge uniformly to f: D --+ lR, or {fn} is said to converge uniformly on D to f, provided that for each positive number E there is an index N such that lf\(x\)fn\(x\)J < E for all indices n ::::: N and all points x in D\. (Patrick Fitzpatrick 264) - Indeed, forE = 1/2 (Patrick Fitzpatrick 265) - taking x = \(3/4\) 1/\(N+l\), (Patrick Fitzpatrick 265) - Definition The sequence of functions {fn : D ---+ IR} is said to be uniformly Cauchy, or {fn} is said to be uniformly Cauchy on D, provided that for each positive number E there is an index N such that lfn+k\(x\)fn\(x\)l < E \(9\.11\) for every index n ~ N, every natural number k, and every point x in D\. (Patrick Fitzpatrick 266) - Theorem 9\.29 The Weierstrass Uniform Convergence Criterion The sequence of functions {fn: D ---+ JR\.} converges uniformly to a function f: D ---+ IR if and only if the sequence {fn: D ---+ JR\.} is uniformly Cauchy (Patrick Fitzpatrick 266) - = lfn+k\(x\)f\(x\) + f\(x\)fn\(x\)l (Patrick Fitzpatrick 266) - Theorem 9\.31 Suppose that {fn: D ---+ JR} is a sequence of continuo us functions that converges uniformly to the function f : D ---+ IR\.\. Then the limit function f : D ---+ JR also is continuous\. (Patrick Fitzpatrick 268) - Theorem 9\.32 Suppose that {fn: [a, b] ----+ JR} is a sequence of integrable functions that converges uniformly to the function f: [a, b] ----+ JR\. Then the limit function f : [a, b] ----+ JR also is integrable\. Moreover, nil,~ [[ fn] = [f\. (Patrick Fitzpatrick 269) - The uniform limit of differentiable functions need not be differentiable \(Exercise 1\)\. (Patrick Fitzpatrick 271) - A function f: I -+ JR, defined on an open interval, is called continuously differentiable provided that it is differentiable and its derivative is continuous\. (Patrick Fitzpatrick 271) - Theorem 9\.33 Let I be an open interval\. Suppose that {fn: I -+ JR} is a sequence of continuously differentiable functions that has the following two properties: i\. The sequence {fn} converges pointwise on I to the function f, and ii\. The derived sequence {f~} converges uniformly on I to the function g\. Then the function f :I -+ lR is continuously differentiable, and f'\(x\) = g\(x\) for all x in I\. (Patrick Fitzpatrick 271) - Theorem 9\.34 Let I be an open interval\. Suppose that {fn: I -+ JR} is a sequence of continuously differentiable functions that has the following two properties: i\. The sequence {fn} converges pointwise on I to the function f, and ii\. The derived sequence {f~} is uniformly Cauchy on I\. (Patrick Fitzpatrick 271) - Then the function f :I -+ IR\. is continuously differentiable, and for each x in I, lim f~\(x\) = f'\(x\) \. (Patrick Fitzpatrick 272) - The Weierstrass Uniform Convergence Criterion (Patrick Fitzpatrick 272) - The principal objective of this section is to show that if the function f : \(r, r\) -+ JR\. is defined by the power series expansion for lxl < r, then f: \(-r, r\) -+ JR\. is differentiable, and moreover, (Patrick Fitzpatrick 275) - The above computation is known as term-by-term differentiation of a series expansion (Patrick Fitzpatrick 275) - The series \.Z::~o ckxk is said to be convergen t uniformly on the set A provided that the sequence of partial sums {sn} converges uniformly on A to the function f\. (Patrick Fitzpatrick 276) - Assume the following: There is a positive number M and a number a with 0 ~ a < 1 such that ~· for all indices k and all x in A \. \(9\.30\) Then the power series \.Z::~o ckxk is uniformly convergent on A\. (Patrick Fitzpatrick 276) - However, the Weierstrass Uniform Convergence Criterion asserts that a sequence of functions converges uniformly if and only if the sequence is uniformly Cauchy (Patrick Fitzpatrick 276) - Moreover, each of the power series and converges uniformly on the interval [ -r, r ] (Patrick Fitzpatrick 277) - Suppose that the nonzero number x 0 is in the domain of convergence of the power series 'E~o ckxk\. Let r be any positive number less than lxo 1\. (Patrick Fitzpatrick 277) - The general result follows by induction since, according to Theorem 9 \.40, the derived series of any power series that converges on \( -r, r\) is another power series that also converges on \( -r, r\)\. (Patrick Fitzpatrick 279) - Then the function f: \(-r, r\) --+ lR has derivatives of all orders (Patrick Fitzpatrick 279) - Since the series 2:::~ 0 ckxk converges at each point between R and r, according to Theorem 9\.40, each of the series and converges uniformly on the interval [R, R]\. (Patrick Fitzpatrick 279) - The above theorem implies that a function defined by a power series expansion on the interval \( -r, r\) coincides with its Taylor series expansion about 0; this is a uniqueness result for the coefficients of a power series expansion (Patrick Fitzpatrick 280) - Weierstrass presented the first example of a continuous function f : IR -+ IR that has the remarkable property that there is no point at which it is differentiable (Patrick Fitzpatrick 283) - We will prove that the sequence of functions {fn} is uniformly Cauchy on IR\. Once this is proven, it follows from the Weierstrass Uniform Convergence Criterion that {fn} converges uniformly on IR\. Then, by Theorem 9\.31, we can conclude that the limit function f, being the uniform limit of a sequence of continuous functions, is continuous (Patrick Fitzpatrick 283)