MATH 201 Exam 1, Fall 2013
American University of Beirut · Calculus and Analytical Geometry III · 7 pages
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Math 201 Quiz 1 Long Sample I
Kamal Aziziheris
Name: ……………………………………….. I.D ...........................................
In computing limits, you may skip the following
1) certain justifications if they are well-known.
2) In LCT, IF you are sure that L=1, write L=1 & skip its proof.
Investigate = Investigate for convergence or divergence
1. (a) Investigate the series
n
n
en
en
7)
1
(
1
(b) Investiagte
1
n
an
given that
n
n
n
n
n
a
a
n
n
5)
cos(
)1
2)(2
3
(
7
|
|
2
1
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Total over 100
(c) Investiagte
1
n
an
given that
n
n
1
n
n
n!
1
|
a| a
d) Investiagate
)
!
(7
sin
1
n
n
n
e) Investigate
2
)
ln
3
ln(1
n
n
f) Investigate
/ 2
3
1
)1
(
n
n
n
(BIG Hint:
n
n
n
e
n
ln
)
2a) Investigate
(3 )!
)1
(
)! 3
(
100
1
n
n
n
n
2a’) Find
n
3n !)
(
3
n
nlim
Hint : Use
|
| a | lim a| a
lim
n
1
n
n
n
n
n
(if this last exists) .
2b) Find the domain of convergence and the domain of Absolute convergence of
2
n
n
n
n
nln n
7
3)
(2x
)1
(
3a) Find the Maclaurin series of
5 3
)
(
x
e
x
f
to deduce
f (150) (0)
....
|
by using the first 3 terms of the appropriat eseries & find |
)
3
2.0
0
5 3
Error
dx
e
Estimate
b
x
4a) (Suppose
n
1
n
a
converges, prove or disprove that
sin(a ))
ln (2
n
1
n
converges
4b) Suppose
n
1
n
a
converges, prove or disprove that
n
n
n
n
a
)
4
)
sin(
(5
)1
(
1
converges
4c) Investigate carefully
n
n
n
n
n
sin( )
)1
(
1
5a) Find
n
n
n
5
1
..
..........
3
1
2
1
1
lim
5b) Use Taylor’s Remainder’s formula to show that
f (x)
= its Maclaurin series
given that
for all x
for all n
e
x
f
x
n
n
&
(1000)
|)
(
|
7
( )
.
(For time limitations, assume x > 0)
5B) Use Taylor’s Remainder formula to find the maximum possible Error (as a simple fraction)
in the approximation of
first 3 terms.
using only the
!
7.0 )
(
0
n
n
n
5c) Use the Maclaurin series of
for all n.
!
to prove
e
n
n n
e x