Math 142A
(P. Fitzsimmons)
Final Exam
December 7, 2001
Solutions
1.
Define of each of the following terms (or statements):
(a) Cauchy sequence (of real numbers)
(b) least upper bound (of a bounded set
S
of real numbers)
(c) radius of convergence (of a power series
∑
∞
n
=0
a
n
x
n
)
(d) inverse function (of a onetoone function
f
)
(e)
f
is continuous at the real number
x
0
2.
Evaluate the indicated limits.
(a)
lim
n
→
+
∞
n
4
n
.
lim
x
→
0
+
x
2
cos(1
/x
)
.
Solution.
(a) Notice that
n
≤
2
n
for all
n
≥
1.
This is clear for
n
= 1; proceeding
inductively, we have
n
+ 1
≤
2
n
+ 1
≤
2
n
+ 2
n
= 2
n
+1
. Therefore,
0
≤
n
4
n
≤
1
2
n
,
which tends to 0 as
n
→ ∞
by a theorem established in class. The limit in question is
therefore 0 by the Squeeze Theorem.
(b) Because

x
2
cos(1
/x
)
 ≤
x
2
and
x
→
x
2
is continuous, the Squeeze Theorem
applies, and we conclude that the limit in question is 0.
3.
Consider the sequence
a
n
=
n
n
2
+ 1
,n
= 0
,
1
,
2
,...
. Find the least upper bound and
the greatest lower bound of
{
a
n
}
, if they exist.
Solution.
Every term of the sequence is
≥
0 and the first (
n
= 0) is equal to 0. Therefore
the G.L.B. is 0. The second term (
n
= 1) is
1
2
, after which the sequence is decreasing.
Therefore the L.U.B. is
1
2
.
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 Fall '01
 Fitzsimmons
 Calculus, Real Numbers, Continuous function, xn + xn

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