PRACTICE PROBLEMS FOR FINAL
The
comprehensive
final will be given on
Wednesday, March 17, 810 am
, in our usual
classroom (MST 124).
It will be a
closedbook test
.
No textbooks, notes, or electronic
devices will be allowed. Scratch paper will be provided to those needing it. You only need
to bring your pens, papers, and erasers.
In preparing for the test, you can practice solving the problems from the list below.
In
addition, take a look at the homework problems (at least one problems on the midterm will
come more or less directly from the homework), and at the examples given in the textbook.
The list below contains problems of different levels of difficulty. Extra hard (in my opinion)
problems are marked by (!).
1.
(!) Prove that, for any positive integer
n
,
2
n
summationdisplay
k
=1
(

1)
k
+1
k
=
2
n
summationdisplay
j
=
n
+1
1
j
.
2.
(!)
Suppose
p
is a polynomial of degree
n
, and
x
0
∈
R
.
Prove that there exist
a
0
,a
1
,...,a
n
∈
R
s.t.
p
(
x
) =
∑
n
k
=0
a
k
(
x

x
0
)
k
for any
x
∈
R
.
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 Winter '08
 staff
 Limit of a sequence, lim xn, Compute lim sup

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