 5.2.5.1.19: Determine the radius of convergence. (Show the details.)
 5.2.5.1.20: Determine the radius of convergence. (Show the details.)
 5.2.5.1.21: Determine the radius of convergence. (Show the details.)
 5.2.5.1.22: Determine the radius of convergence. (Show the details.)
 5.2.5.1.23: Determine the radius of convergence. (Show the details.)
 5.2.5.1.24: Determine the radius of convergence. (Show the details.)
 5.2.5.1.25: Determine the radius of convergence. (Show the details.)
 5.2.5.1.26: Determine the radius of convergence. (Show the details.)
 5.2.5.1.27: Determine the radius of convergence. (Show the details.)
 5.2.5.1.28: Determine the radius of convergence. (Show the details.)
 5.2.5.1.29: Determine the radius of convergence. (Show the details.)
 5.2.5.1.30: Determine the radius of convergence. (Show the details.)
 5.2.5.1.31: Thi~ is often convenient or nece~sary in the power series method. S...
 5.2.5.1.32: Thi~ is often convenient or nece~sary in the power series method. S...
 5.2.5.1.33: Thi~ is often convenient or nece~sary in the power series method. S...
 5.2.5.1.34: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.35: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.36: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.37: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.38: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.39: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.40: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.41: Find a power series solution in powers of x. (Show the details of y...
 5.2.5.1.42: TEAM PROJECT. Properties from Power Series. In the next sections we...
 5.2.5.1.43: CAS EXPERIMENT. Information from Graphs of Partial Sums. In connect...
Solutions for Chapter 5.2: Theory of the Power Series Method
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 5.2: Theory of the Power Series Method
Get Full SolutionsSince 25 problems in chapter 5.2: Theory of the Power Series Method have been answered, more than 46718 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: Theory of the Power Series Method includes 25 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.