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Math 20D  Spring 2006  Midterm 2 Prof. Mike Zabrocki
score
1
24
2
25
3
25
4
26
BONUS
10
total
100
Name:
Time of section:
1. Determine if the following sums diverge, converge absolutely or converge conditionally. State which
test you use to justify your answer and the conclusion which follows from that test.
(a)
∑
∞
n
=1
sin(
1
n
2
)
.
(b)
∑
∞
n
=2
(

1)
n
√
n

n
2
Since sin(
1
n
2
) =
1
n
2

(
1
n
2
)
3
3!
+
· · ·
,
This is an alternating series and
1
n
2

√
n
we know that lim
n
→∞
sin
(
1
n
2
)
1
n
2
= 1
is positive and decreasing and lim
n
→∞
1
n
2

√
n
= 0
and by limit comparison test we know that
Hence by A.S.T. converges conditionally.
∑
n
≥
0
1
n
2
converges
since lim
n
→∞
1
n
2

√
n
1
n
2
= 1
implies that
∑
n
≥
1
sin(
1
n
2
) converges.
then this series converges absolutely
The series is positive hence converges
by the L.C.T. with
∑
1
n
2
absolutely
(c)
∑
∞
n
=1
sin(
πn
2
)
1+sin(
n
2
)
(d)
∑
∞
n
=1
(1

e
1
/n
)
lim
n
→∞
1
1+sin(
n
2
)
6
= 0
Since 1

e
1
n
= 1

(1 +
1
n
+
(
1
n
)
2
2!
+
(
1
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This note was uploaded on 06/12/2008 for the course MATH 20D taught by Professor Mohanty during the Spring '06 term at UCSD.
 Spring '06
 Mohanty
 Math

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