 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.1: What are power series? Why are these series very important in compl...
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: (Powers missing) Show that if ~ a"z'YI has radius of convergence R ...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.2: State from memory the ratio test, the root test, and the CauchyHad...
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: (Convergence behavior) Illustrate the facts shown by Examples 13 b...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.3: What is absolute convergence? Conditional convergence? Uniform conv...
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.4: What do you know about the convergence of power series?
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.5: What is a Taylor series? What was the idea of obtaining it from Cau...
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.6: Give examples of practical methods for obtaining Taylor series.
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.7: What have power series to do with analytic functions?
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Prove that the given series converges uniformly In the indicated re...
 15.8: Can propel1ies of functions be discovered from their Maclaurin seri...
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.9: Make a list of Maclaurin series of c. cos z. sin z, cosh z, sinh z,...
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.10: What do you know about adding and multiplying power series?
 15.15.1: Are the following sequences Zl, Z2, ... , Zn> ... bounded? Converge...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: Find the radius of convergence in two ways: (a) directly by the Cau...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.11: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: Illustrate Theorem 1 by an example of your own.
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: (Addition and subtraction) Write our the details of the proof on te...
 15.15.4: Find the Taylor or Maclaurin series of the given function with the ...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.12: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: (Uniqueness of limit) Show that if a sequence converges. its limit ...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: (Cauchy product) Show that (1  Z)2 = L';;~O (n + l)zn tal by usin...
 15.15.4: Find the Maclaurin series by tennwise integrating the integrand. (T...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.13: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: (Addition) If ZI, Z2, ... converges with the limit [and ZI *, Z2 *,...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: (Cauchy product) Show that the Cauchy product of L~~O zn/n! multipl...
 15.15.4: Find the Maclaurin series by tennwise integrating the integrand. (T...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.14: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: (Multiplication) Show that under the assumptions of Prob. L3 the se...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: (On Theorem 3) Prove that Vn ~ I as n ~ ex; (as claimed in the proo...
 15.15.4: Find the Maclaurin series by tennwise integrating the integrand. (T...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.15: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: (Boundedness) Show that a complex sequence is bounded if and onl y ...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: (On Theorems 3 and 4) Find further examples of your own.
 15.15.4: Find the Maclaurin series by tennwise integrating the integrand. (T...
 15.15.5: Find the region of uniform convergence. (Give reason.)
 15.16: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)6...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: State clearly and explicitly where and how you are using Theorem 2....
 15.15.4: CAS PROJECT. sec, tan, arcsin. (a) Euler numbers. The Maclaurin ser...
 15.15.5: CAS PROJECT. Graphs of Partial Sums. (a) Figure 365. Produce this e...
 15.17: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)~...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: State clearly and explicitly where and how you are using Theorem 2....
 15.15.4: (Inverse sine) Developing uV I  Z2 and integrating, show that arcs...
 15.15.5: TEAM PROJECT. Uniform Convergence. (a) Weierstrass Mtest. Give a p...
 15.18: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: Are the following series convergent or divergent? (Give a reason.) ...
 15.15.2: Find the center and the radius of convergence of the following powe...
 15.15.3: State clearly and explicitly where and how you are using Theorem 2....
 15.15.4: (Undetennined coefficients) Using the relation sin z = tan Z cos Z ...
 15.15.5: Show that (9) in Sec. 12.5 with coefficients (10) is a solution of ...
 15.19: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)2...
 15.15.2: CAS PROJECT. Radius of Convergence. Write a program for computing R...
 15.15.3: State clearly and explicitly where and how you are using Theorem 2....
 15.15.4: TEAM PROJECT. Properties from Maclaurin Series. Clearly, from serie...
 15.15.5: Show that (9) in Sec. 12.5 with coefficients (10) is a solution of ...
 15.20: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)....
 15.15.2: TEAM PROJECT. Radius of Convergence. (a) Formula (6) for R contains...
 15.15.3: State clearly and explicitly where and how you are using Theorem 2....
 15.21: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)
 15.22: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)2...
 15.23: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)....
 15.24: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: Are the following series convergent or divergent? (Give a reason.)
 15.25: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: What is the difference between (7) and just stating IZn+l/Znl < I?
 15.26: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: Illustrate Theorem 2 by an example of your choice.
 15.27: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: For what n do we obtain the term of greatest absolute value of the ...
 15.28: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: Give another example showing that Theorem 7 is more general than Th...
 15.29: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: CAS PROJECT. Sequences and Series. (a) Write a program for graphing...
 15.30: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1: TEAM PROJECT. Series. ta) Absolute convergence. Show that if a seri...
 15.31: Does every function fez) have a Taylor series?
 15.32: Does there exist a Taylor series in powers of z  1  i that diverg...
 15.33: Do we obtain an analytic function if we replace x by z in the Macla...
 15.34: Using Maclaurin series. show that if fez) is even. its integral (wi...
 15.35: Obtain the first few terms of the Maclaurin series of tan z by usin...
Solutions for Chapter 15: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 15
Get Full SolutionsSince 145 problems in chapter Chapter 15 have been answered, more than 66256 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter Chapter 15 includes 145 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.