 15.15.1.111: What are power series? Why are these series very important in compl...
 15.15.1.112: State from memory the ratio test, the root test, and the CauchyHad...
 15.15.1.113: What is absolute convergence? Conditional convergence? Uniform conv...
 15.15.1.114: What do you know about the convergence of power series?
 15.15.1.115: What is a Taylor series? What was the idea of obtaining it from Cau...
 15.15.1.116: Give examples of practical methods for obtaining Taylor series.
 15.15.1.117: What have power series to do with analytic functions?
 15.15.1.118: Can propel1ies of functions be discovered from their Maclaurin seri...
 15.15.1.119: Make a list of Maclaurin series of c. cos z. sin z, cosh z, sinh z,...
 15.15.1.120: What do you know about adding and multiplying power series?
 15.15.1.121: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.122: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.123: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.124: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.125: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.126: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.127: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.128: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.129: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.130: Find the radius of convergence. Can you identify the sum as a famil...
 15.15.1.131: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.132: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.133: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.134: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.135: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.136: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.137: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.138: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.139: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.140: Find the Taylor or Maclaurin series with the given point as center ...
 15.15.1.141: Does every function fez) have a Taylor series?
 15.15.1.142: Does there exist a Taylor series in powers of z  1  i that diverg...
 15.15.1.143: Do we obtain an analytic function if we replace x by z in the Macla...
 15.15.1.144: Using Maclaurin series. show that if fez) is even. its integral (wi...
 15.15.1.145: Obtain the first few terms of the Maclaurin series of tan z by usin...
Solutions for Chapter 15: Power Series, Taylor Series
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 15: Power Series, Taylor Series
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions. Since 35 problems in chapter 15: Power Series, Taylor Series have been answered, more than 49903 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter 15: Power Series, Taylor Series includes 35 full stepbystep solutions.

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

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).