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m201pt3.docMATH 201-01 Practice Exam III
1. Find the radius of convergence of the power series
a.
å
¥
=
-
0
2
4
)
1
(
n
n
n
n
x
n
b.å
¥
=
1
)
3
(
n
n
n
x
c.å
¥
=
0
)!
2
(
!
n
n
n
x
n
2. Find the interval of convergence for each series
a.
å
¥
=
+
-
-
0
2
)
1
(
)
2
(
)
1
(
n
n
n
n
x
b.å
¥
=
-
1
)
2
(
3
n
n
n
n
x
c.å
¥
=
-
0
2
!
2
)
3
(
n
n
n
n
n
x
3. Use known power series representation to find a power series representation of the given function centered at c.
a.
2
,
1
2
2
)
(
=
-
=
c
x
x
f
b.0
,
arctan
)
(
=
=
c
x
x
f
c.0
,
3
2
)
(
=
+
=
c
x
x
x
f
4. (a) Find a power series representation for
2
)
2
1
(
1
)
(
x
x
f
+
=
(b) Use it to find the sum of the infinite series
å
¥
=
-
+
-
1
1
2
1
2
3
)
1
(
n
n
n
n
5. Use known Maclaurin series to find the power series for the function
a.
x
x
f
2
cos
)
(
=
b.2
/
2
)
(
x
e
x
f
-
=
6. Use a power series to approximate the value of the integral
dx
x
x
ò
-
1
0
2
cos
1
with the error of less than 0.00017. Evaluate each improper integral if it converges.
a.
dx
x
ò
¥
1
2
2
b.dx
x
ò
1
0
3
/
2
2
c.dx
e
x
ò
¥
-
0
2
