solnsexamII - 1 SOLUTIONS(1(20 points(a If the sequence an...

1. SOLUTIONS
(1)(20 points)
( a ) If the sequence a
n
satisfies lim
n →∞
a
n +1
- a
n
= 0 then lim
n →∞
a
n
exists and is finite.
F
Take a
n
=
√
n.
( b ) If q is a polynomial of degree 5 and q (0) = q (0) = q
(4)
(0) = 0 and q (0) = 1 , q (0) =
-
1
6
, q
(5)
(0) =
1
120
then q is the fifth degree Taylor polynomial for f ( x ) = sin x at c = 0 .
T
Since f (0) = f (0) = f (4) (0) = 0 and f (0) = 1 , f (0) = - 1
6
, f (5) (0) = 1
120
.
( c ) The matrix
A =





0 1 0
0 0 1
0 0 0





is diagonalizable.
F
One way to see this is that the only eigenvalue is 0 , it has multiplicity 3 , and the eigenspace
N ( A - 0 I ) = { ( t, 0 , 0) T : t ∈ R } so there are not three linearly independent eigenvectors.
Another way to see this is that if A were diagonalizable, then for some invertible matrix X
we would have
A = XDX
- 1
and D is the diagonal matrix with the eigenvalues of A on its diagonal. But this would be the
0 matrix and as a result A is also the 0 matrix, but it is not, so A can not be diagonalizable.
1

2
( d ) If an eigenvalue of the matrix A has multiplicity greater than one, then A is defective.
F
Defective means not diagonalizable, but the identity matrix has sole eigenvalue 1 and is
already diagonal.