Math 1B
Exam 3 Review
The third exam will be on Friday, May 13th. It will cover sections 11.1 – 11.10. You are allowed to use
one 3” by 5” notecard (no specific examples, known Maclaurin series on pg. 762 are ok). Prepare well
and good luck!
You will need to be able to:
1.
2.
3.
4.
5.
6.
7.
8.
Determine whether a sequence is convergent or divergent.
Determine whether a series converges or diverges using the following tests:
• The Test for Divergence
• The Integral Test
• The Comparison Test
• The Limit Comparison Test
• The Alternating Series Test
• The Ratio Test
• The Root Test
Determine whether a series is absolutely convergent, conditionally convergent, or divergent.
Find the sum of convergent geometric series, telescoping series, and Taylor series.
Find the radius and interval of convergence of a power series.
Differentiate and integrate power series.
Find the Taylor series or Maclaurin series of a function (and its radius of convergence).
Use a known Maclaurin series to find the Maclaurin series of a function.
Check out the Chapter 11 Review Exercises, pages 779-780, # 1-7 odd, 11-31 odd, 41-51 odd, 55
Additional Problems:
1.
Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.
a)
b)
€
c)
€
d)
€
2.
€
€
) # π &,∞
*cos% (+ $ n '. n=1
ln n
an =
n
∞
(" 4 % n +
)$1+ ' ,
*# n & - n=1
# 9 − n2 &
$
'
% n (2n + 3) (
Determine whether each series is convergent or divergent. If it converges, find the sum.
∞
a)
n2
∑ 4n 2 −1
n=1
∞
b)
1
∑ n(n + 2)
n=1
3n + (−4 )
c) ∑
6n
n=1
∞
€
n
∞
d)
∑ [tan (n + 1) − tan n]
−1
−1
n=1
€
€
€
3.
€
16 64
−
+ ...
3 9
∞
(n −1)( n + 1)
f) ∑
10n 2
n=1
e) 3 − 4 +
Determine whether each series is convergent or divergent.
∞
a)
n=1
∞
b)
∑
n= 0
€
€
€
€
€
€
4.
€
+ 4)
n
Determine whether each of the following series is absolutely convergent, conditionally
convergent, or divergent.
b)
€
2
∑n
(−1)
n
∑ n ln n
n= 2
∞
€
(3n
2n
sin 2 n
n
n=1
2
∞
n!)
(
d) ∑
n= 0 (2n )!
∞
n 2 − 5n
e) ∑ 3
n=1 n + n + 1
∞
1
f) ∑
n+4
n=1
∞
4n
g) ∑
n
n= 0 n + 6
a)
€
(2n −1)
∞
c)
∞
€
2 −n 3
∑n e
∑
n
(−1) (n + 1)3n
2 2n +1
n2
n
c) ∑ (−1) 3
n +8
n=1
∞
n
n
d) ∑
n=1 n!
n=1
∞
5.
Find the radius of convergence and interval of convergence for each series.
∞
a)
∑ (−1)
n=1
∞
€
n
xn
n 2 5n
n
2 n ( x − 3)
b) ∑
n+3
n= 0
n
n
∞
2 ( x − 2)
c) ∑
n=1 ( n + 2)!
€
6.
€
Find a power series representation for f (x) =
x2
(1− x )
2
. State the radius of convergence.
7.
Find a power series representation for f (x) = ln(4 − x) . State the radius of convergence.
8.
Find the first five nonzero terms of the Taylor series for f (x) = x about a = 1.
9.
1
Find the first five nonzero terms of the Taylor series for f (x) = about a = −3 .
x
€
€
10.
Use a known Maclaurin series to find the Maclaurin series for each function and its radius of
€
convergence. Simplify your answer.
€
€
a) f (x) = cos x
x2
b) f (x) =
1+ x
c) f (x) = xe 2x
€
€
Answers to Additional Problems
1.
a) Converges to 1
b) Converges to 0
c) Converges to e 4
1
d) Converges to −
2
2.
€
a) Diverges (Test for Divergence)
b) Telescoping
series: converges to
€
c) Geometric series: converges to
€
€
3
5
3
4
d) Telescoping series: converges to
e) Geometric series: diverges
f) Diverges (Test for Divergence)
3.
π
4
€
a) Converges (Integral Test)
b) Diverges (Root Test)
c) Converges (Comparison Test w/
d) Converges (Ratio Test)
1
3 )
n2
1
e) Diverges (Limit Comparison Test w/ )
n
€
f) Diverges (Integral Test)
n
&4#
g) Converges (Comparison Test w/ $ ! )
%6"
€
4.
a) Conditionally convergent (Alternating Series Test, Integral Test)
b) Absolutely convergent (Ratio Test)
c) Conditionally convergent (Alternating Series Test, Limit Comparison or Integral Test)
d) Diverges (Ratio Test)
5.
a) R = 5 , I = [−5,5]
"5 7 %
1
b) R = , I = $ , '
#2 2 &
2
c) R = ∞ , I = (−∞,∞)
€
6.
€ nx n+1
∑
n=1
€
7.
ln 4 − ∑
8.
1+
9.
1 1
1
1
1
2
3
4
− − ( x + 3) − ( x + 3) − ( x + 3) −
( x + 3)
3 9
27
81
243
10.
a)
∞
R=1
∞
€
xn
n
n=1 n4
1
1
1
5
2
3
4
( x −1) − ( x −1) + ( x −1) − ( x −1)
2
8
16
128
∞
€
R=4
xn
n
∑(−1) (2n)!
R=∞
n=0
∞
b)
∑(−1)
n
x n+2
R=1
n=0
∞
c)
€
2 n x n +1
n!
n= 0
∑
R=∞