 9.4.1: Determine whether the sequence is convergent or divergent. If it is...
 9.4.2: Determine whether the sequence is convergent or divergent. If it is...
 9.4.3: Determine whether the sequence is convergent or divergent. If it is...
 9.4.4: Determine whether the sequence is convergent or divergent. If it is...
 9.4.5: Determine whether the sequence is convergent or divergent. If it is...
 9.4.6: Determine whether the sequence is convergent or divergent. If it is...
 9.4.7: Determine whether the sequence is convergent or divergent. If it is...
 9.4.8: A sequence is deined recursively by the equations a1 1, an11 1 3 sa...
 9.4.9: Show that limn l ` n 4 e 2n 0 and use a graph to ind the smallest v...
 9.4.10: Determine whether the series is convergent or divergent
 9.4.11: Determine whether the series is convergent or divergent
 9.4.12: Determine whether the series is convergent or divergent
 9.4.13: Determine whether the series is convergent or divergent
 9.4.14: Determine whether the series is convergent or divergent
 9.4.15: Determine whether the series is convergent or divergent
 9.4.16: Determine whether the series is convergent or divergent
 9.4.17: Determine whether the series is convergent or divergent
 9.4.18: Determine whether the series is convergent or divergent
 9.4.19: Determine whether the series is convergent or divergent
 9.4.20: Determine whether the series is convergent or divergent
 9.4.21: Determine whether the series is convergent or divergent
 9.4.22: Determine whether the series is conditionally convergent, absolutel...
 9.4.23: Determine whether the series is conditionally convergent, absolutel...
 9.4.24: Determine whether the series is conditionally convergent, absolutel...
 9.4.25: Determine whether the series is conditionally convergent, absolutel...
 9.4.26: Find the sum of the series
 9.4.27: Find the sum of the series
 9.4.28: Find the sum of the series
 9.4.29: Find the sum of the series
 9.4.30: Find the sum of the series
 9.4.31: Express the repeating decimal 4.17326326326 . . . as a fraction.
 9.4.32: Show that cosh x > 1 1 1 2 x 2 for all x.
 9.4.33: For what values of x does the series o ` n1 sln xd n converge?
 9.4.34: Find the sum of the series o ` n1 s21d n11 n 5 correct to four deci...
 9.4.35: (a) Find the partial sum s5 of the series o ` n1 1yn 6 and estimate...
 9.4.36: Use the sum of the irst eight terms to approximate the sum of the s...
 9.4.37: (a) Show that the series o ` n1 n n s2nd! is convergent. (b) Deduce...
 9.4.38: Prove that if the series o ` n1 an is absolutely convergent, then t...
 9.4.39: Find the radius of convergence and interval of convergence of the s...
 9.4.40: Find the radius of convergence and interval of convergence of the s...
 9.4.41: Find the radius of convergence and interval of convergence of the s...
 9.4.42: Find the radius of convergence and interval of convergence of the s...
 9.4.43: Find the radius of convergence of the series o ` n1 s2nd! sn!d 2 x n
 9.4.44: Find the Taylor series of fsxd sin x at a y6.
 9.4.45: Find the Taylor series of fsxd cos x at a y3.
 9.4.46: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.47: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.48: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.49: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.50: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.51: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.52: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.53: Find the Maclaurin series for f and its radius of convergence. You ...
 9.4.54: Evaluate y e x x dx as an ininite series
 9.4.55: Use series to approximate y 1 0 s1 1 x 4 dx correct to two decimal ...
 9.4.56: (a) Approximate f by a Taylor polynomial with degree n at the numbe...
Solutions for Chapter 9.4: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 9.4
Get Full SolutionsChapter 9.4 includes 56 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Since 56 problems in chapter 9.4 have been answered, more than 57597 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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 equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

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

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

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·