 5.1.1: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.2: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.3: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.4: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.5: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.6: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.7: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.8: In each of 1 through 8 determine the radius of convergence of the g...
 5.1.9: In each of 9 through 16 determine the Taylor series about the point...
 5.1.10: In each of 9 through 16 determine the Taylor series about the point...
 5.1.11: In each of 9 through 16 determine the Taylor series about the point...
 5.1.12: In each of 9 through 16 determine the Taylor series about the point...
 5.1.13: In each of 9 through 16 determine the Taylor series about the point...
 5.1.14: In each of 9 through 16 determine the Taylor series about the point...
 5.1.15: In each of 9 through 16 determine the Taylor series about the point...
 5.1.16: In each of 9 through 16 determine the Taylor series about the point...
 5.1.17: Given that y = n=0 nxn, compute y and y and write out the first fou...
 5.1.18: Given that y = n=0 anxn, compute y and y and write out the first fo...
 5.1.19: In each of 19 and 20 verify the given equation.
 5.1.20: In each of 19 and 20 verify the given equation.
 5.1.21: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.22: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.23: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.24: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.25: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.26: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.27: In each of 21 through 27 rewrite the given expression as a sum whos...
 5.1.28: Determine the an so that the equation n=1 nanxn1 + 2 n=0 anxn = 0 i...
Solutions for Chapter 5.1: Review of Power Series
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 5.1: Review of Power Series
Get Full SolutionsSince 28 problems in chapter 5.1: Review of Power Series have been answered, more than 14459 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Chapter 5.1: Review of Power Series includes 28 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

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.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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).

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·