Math 1B: Discussion Exercises
GSI: Theo Johnson-Freyd
http://math.berkeley.edu/~theojf/09Summer1B/
Find two or three classmates and a few feet of chalkboard. As a group, try your hand at the
following exercises. Be sure to discuss how to solve the exercises — how you get the solution is
much more important than whether you get the solution. If as a group you agree that you all
understand a certain type of exercise, move on to later problems. You are not expected to solve all
the exercises: some are very hard.
Exercises marked with an §are from Single Variable Calculus: Early Transcendentals for UC
Berkeley by James Stewart. Others are my own or are independently marked.
Alternating Series and Absolute Convergence
Let bnbe a positive decreasing sequence: bn≥bn+1 ≥0 for every n. Then P(−1)nbnconverges.
Moreover, if we truncate the series after then Nth term, and estimate P∞
0(−1)nbnby sN=
PN
0(−1)nbn, then the error of the estimate is at most |bN+1|.
A series Panconverges absolutely if P|an|converges. It is a theorem that if Panis abso-
lutely convergent, then it is convergent. But many series are convergent without being absolutely
convergent. For example, the Alternating Harmonic series P∞
1(−1)n−1/n converges by the alter-
nating series test, but P∞
1(−1)n−1/n=P∞
11/n is the divergent Harmonic series. A series that
converges but does not absolutely converge is said to converge conditionally.
1. For what values of pdoes
∞
X
n=1
(−1)n−1
np
(a) converge absolutely?
(b) converge conditionally?
(c) diverge?
2. For what values of rdoes
∞
X
n=1
rn
(a) converge absolutely?
(b) converge conditionally?
(c) diverge?
3. §Determine whether the following series are absolutely convergent, conditionally convergent,
or divergent:
(a)
∞
X
1
(−1)nn
n+ 2 (b)
∞
X
2
(−1)n
√n(c)
∞
X
1
(−1)n−1
ln(n+ 4)
(d)
∞
X
1
(−1)nn
√n3+ 2 (e)
∞
X
1
(−1)n
10n(f)
∞
X
1
cos πn
n2
(g)
∞
X
1
(−1)n(−1)ncos π
n(h)
∞
X
1−n
5n(i)
∞
X
1−1
2nn
1
