Written Assignment 5
esolutions
determine_the_radius_q.rtfWA 5, p. 1
Name:
College ID:
Thomas Edison State College
Calculus II (MAT-232)
Section no.:
Semester and year:
Written Assignment 5
Answer all assigned exercises, and show all work. Each exercise is worth 5 points.
*Submitting a graph is not required; however, you are encouraged to create one for your own benefit and to include (or describe) one if possible.
S ection 8 . 6
4 . Determine the radius and interval of convergence.
0
2
k
k
k
k
x
¥
=
å
10 . Determine the radius and interval of convergence.
2
4
1
(32)
k
k
x
k
¥
=
+
å
12 . Determine the radius and interval of convergence.
1
(1)
(31)
k
k
k
x
k
¥
=
-
-
å
16 . Determine the radius and interval of convergence.
2
21
2
(!)
(2)!
k
k
k
x
k
¥
+
=
å
24 . Determine the interval of convergence and the function to which the given power series converges.
0
3
4
k
k
x
¥
=
æö
ç÷
èø
å
26 . Find a power series representation of f(x) about c = 0 (refer to example 6.6). Also, determine the radius and interval of convergence, and graph f(x) together with the partial sums
3
0
k
k
k
ax
=
å
and6
0
k
k
k
ax
=
å
.
3
()
1
fx
x
=
-
28 . Find a power series representation of f(x) about c = 0 (refer to example 6.6). Also, determine the radius and interval of convergence, and graph f(x) together with the partial sums
3
0
k
k
k
ax
=
å
and6
0
k
k
k
ax
=
å
.
2
2
()
1
fx
x
=
-
S ection 8 . 7
4 . Find the Maclaurin series (i.e., Taylor series about c = 0) and its interval of convergence.
()cos2
fxx
=
6 . Find the Maclaurin series (i.e., Taylor series about c = 0) and its interval of convergence.
()
x
fxe
-
=
1 0 . Find the Taylor series about the indicated center, and d etermine the interval of convergence.
()cos,/2
fxxc
p
==-
14 . Find the Taylor series about the indicated center, and determine the interval of convergence.
1
(),0
5
fxc
x
==
+
22 . Prove that the Taylor series converges to f(x) by showing that
()0as
n
Rxn
®®¥
.
2
0
cos(1)
(2)!
k
k
k
x
x
k
¥
=
=-
å
24 . Prove that the Taylor series converges to f(x) by showing that
()0as
n
Rxn
®®¥
.
0
(1)
!
k
xk
k
x
e
k
¥
-
=
=-
å
30 . Use a known Taylor series to find the Taylor series about c = 0 for the given function, and find its radius of convergence.
1
()
x
e
fx
x
-
=
S ection 8 . 8
4 . Use an appropriate Taylor series to approximate the given value, accurate to within
11
10
-
.
cos3.04
8. Use a known Taylor series to conjecture the value of the limit.
22
6
0
sin
lim
x
xx
x
®
-
1 2 . Use a known Taylor series to conjecture the value of the limit.
2
0
1
lim
x
x
e
x
-
®
-
1 6 . Use a known Taylor polynomial with n nonzero terms to est imate the value of the integral.
1
1
0
tan,5
xdxn
-
=
ò
1 8 . Use a known Taylor polynomial with n nonzero terms to estimate the value of the integral.
1
0
,4
x
edxn
=
ò
2 4 . Us e the Binomial Theorem to find the first five terms of the Maclaurin series.
3
()12
fxx
=+
