Math 20D
Final Exam
96 points
8AM Friday June 11, 2004
•
Print Name, ID number and Section on your blue book.
•
BOOKS and CALCULATORS are NOT allowed.
Both sides of one page of NOTES is allowed.
•
You must show your work to receive credit.
1. (10 points) It can be shown that

sin(
x

1)

+

sin
x

>
1
/
2 for all
x
. Using this fact,
or otherwise, determine which of the following series converge and give reasons for
your answers.
(a)
∞
s
n
=1
2

sin
n

n
2
(b)
∞
s
n
=1
2

sin
n

n
2. (12 points) Consider the series
∞
s
n
=0
(1

x
)
n
2
n
+ 1
.
(a) What is its radius of convergence?
(b) For what values of
x
does it converge conditionally?
(c) For what values of
x
does it converge absolutely?
3. (9 points per equation) Solve the following di±erential equations. If initial conditions
are given, ²nd the particular solution. If no initial conditions are given, ²nd the
general solution.
(a)
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 Fall '06
 Mohanty
 Math, Differential Equations, Equations, initial conditions, blue book

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