MATH 1352-H02
Exam III
April 16, 2002
Answer the problems on separate paper. You do not need to rewrite the problem statements on
your answer sheets. Do your own work. Show all relevant steps which lead to your solutions.
Retain this question sheet for your records.
1.
Determine which sequences {an} converge and which diverge. Find the limit of each
convergent sequence. In each case, provide reasons for your conclusion. Do three (3)
of the following problems
a.
c.
2.
an '
n 2 &2 n &1
n%1
b.
d.
n&1
an '
an '
n
2 n % sin n
n & cos 5 n
Find the sum of the following infinite series, if it converges. Provide reasons for your
conclusion.
4
a.
3.
(&1) n
n
an ' 1 %
j
n'0
5
2n
&
3
4
n
Determine which of the following infinite series converge and which diverge. In each
case, provide reasons for your conclusion. Do four (4) of the following problems
4
a.
j
n'1
4
c.
j
n'1
4
e.
j
n'2
1
4
b.
n2 % 1
2n
(2 n % 1)!
1
n ln n
j
n'1
4
d.
j
n'1
10 n % 1
n (n % 1) (2 n & 1)
en
n3
4.
Determine which of the following infinite series converge absolutely, which converge
conditionally and which diverge. In each case, provide reasons for your conclusion. Do
two (2) of the following problems
4
a.
n %1
j (&1)
n'1
4
c.
5.
n %1
j (&1)
4
n % 11
2n % 7
n %1
j (&1)
b.
n'1
n
(n % 2) 2
n%1
n'1
2n
It can be shown, using powers series, that
4
sin
12
' j
n'0
(&1) n
(2 n % 1)!
2n%1
12
'
Use the above series to find the value of sin
reasons for your conclusion.
12
12
&
1
3!
3
12
%
1
5!
5
12
&
1
7!
7
12
% ã.
to six decimal places accuracies. Provide