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Math 140A Test 2
100 points
November 23, 2009
Professor Evans
Directions:
Show all work. In your proofs, state where you use the hypotheses.
Notation:
Throughout,
z
and
a
n
(
n
= 0
,
1
,
2
,
3
,...
) are complex. (If you can only handle the
real case, you still get considerable partial credit.)
Points:
Each problem is worth 20 points.
(1)
Given that
a
n
→
0, show that (
a
1
+
···
+
a
n
)
/n
→
0, as
n
→ ∞
.
Solution: Let
± >
0. Let
s
n
=
a
1
+
···
+
a
n
. There exists an integer
N
such that

a
n

< ±
for
all
n > N
. There exists a positive constant
M
(independent of
±
) for which

a
n

< M
for all
n
≤
N
. Thus by the triangle inequality,

s
n
 ≤
NM
+ (
n

N
)
±
. Thus

s
n

/n < NM/n
+
±
, so

s
n

/n <
2
±
for all
n > NM/±
. This shows the desired result that
s
n
/n
→
0.
(2)
Consider the power series
∑
∞
n
=0
z
n
/
(
n
+ 1). Determine the values of
z
for which the series
(A)
diverges
(B)
converges. Justify brieﬂy.
Solution: It’s easy to see using the ratio test or the root test that the series diverges for
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This note was uploaded on 09/30/2011 for the course MATH 140a taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
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 Math

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