San José City College
Mathematics 72
Practice Exercises for the Final Examination
1.
Determine whether the series
9
8
2
3
2
5
!
k
k
k
+
=
∞
∑
converges or diverges.
Explain your reasoning.
2.
Determine whether the series
1
1
sin
n
n
∞
=
⎛
⎞
⎜
⎟
⎝
⎠
∑
converges or diverges.
Explain your reasoning.
3.
Determine whether the series
3
1
1
4
5
n
n
n
∞
=
+
+
∑
converges or diverges.
Explain your reasoning.
4.
Let
and
1
1
a
=
1
sin( )
n
n
n
a
a
n
+
=
1
n
≥
,
for
.
Use the Ratio Test to show that
converges.
1
n
n
a
∞
=
∑
5.
Determine whether the series
5
1
5
n
n
n
∞
=
∑
converges or diverges.
Explain your reasoning.
6.
Using the Taylor series for
( )
x
f x
e
=
,
obtain a power series representation for
3
( )
x
g x
xe
−
=
and
for
1
( )
1
x
h x
e
x
=
−
−
.
7.
Find the sum of the following series:
5
7
8
1
2
k
k
k
+
=
∞
∑
.

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