Syed Shah
Assignment 3
MATH262, Fall 2009
due 10/18/2009 at 11:59pm EDT.
You may attempt each problem a maximum of 7 times.
1.
(1 pt) The function
f
(
x
) =
1
(
1

10
x
)
2
is represented as a
power series
f
(
x
) =
∞
∑
n
=
0
c
n
x
n
.
Find the first few coefficients in the power series.
c
0
=
c
1
=
c
2
=
c
3
=
c
4
=
Find the radius of convergence
R
of the series.
R
=
.
Correct Answers:
•
1
•
20
•
300
•
4000
•
50000
•
0.1
2.
(1 pt) The function
f
(
x
) =
6
x
arctan
(
5
x
)
is represented as
a power series
f
(
x
) =
∞
∑
n
=
0
c
n
x
n
.
Find the first few coefficients in the power series.
c
0
=
c
1
=
c
2
=
c
3
=
c
4
=
Find the radius of convergence
R
of the series.
R
=
.
Correct Answers:
•
0
•
0
•
30
•
0
•
250
•
0.2
3.
(1 pt) Compute the 7th derivative of
f
(
x
) =
cos
(
4
x
3
)

1
x
5
at
x
=
0.
f
(
7
)
(
0
) =
Hint: Use the MacLaurin series for
f
(
x
)
.
Correct Answers:
•
53760
4.
(1 pt) Find the degree 3 Taylor polynomial
T
3
(
x
)
of func
tion
f
(
x
) = (

3
x
+
15
)
3
/
2
at
a
=
2.
T
3
(
x
) =
Correct Answers:
•
27 + 13.5 * (x2) + 2.25/2 * (x2)**2 + 0.375/6 * (x2)**3
5.
(1 pt) The Taylor series for
f
(
x
) =
ln
(
sec
(
x
))
at
a
=
0 is
∞
∑
n
=
0
c
n
x
n
.
Find the first few coefficients.
c
0
=
c
1
=
c
2
=
c
3
=
c
4
=
Find the exact error in approximating ln
(
sec
(
0
.
3
))
by its fourth
degree Taylor polynomial at
a
=
0
.
The error is
Correct Answers:
•
0
•
0
•
0.5
•
0
•
0.0833333333333333
•
1.66559260579585E05
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 Spring '08
 FABER
 Power Series, Taylor Series

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