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.
Sequences
Asequence is an infinite list of numbers. In this class, and in most of math, we generally begin
at 0: the beginning of a sequence is called the “zeroth” term, and then there’s the “first” term, then
the “second”, etc. But really the indexing — is the sequence {a0, a1, a2, . . . }or {a4, a5, a6, . . . }—
is largely irrelevant. In any case, the formal definition is that “A sequence is a function from the
non-negative integers to the real numbers.”
A sequence might or might not converge to a number, meaning it gets closer and closer to that
number. A definition: limn→∞ an=Lif |L−an|is small for sufficiently large n, i.e. if for any
given > 0 there exists an Nsuch that for all n>N, > |L−an|.
Whether, and to what, a sequence converges does not depend on the “early” values (or indeed
on any given term) of a sequence. If anand bneach converge, then so do an+bnand anbn, and
the limits add and multiply correctly. If f(x) is a continuous function and anconverges to some
number a∞, then f(an) converges to f(a∞).
If a sequence is monotonic — strictly increasing or strictly decreasing — and bounded, then it
converges.
A sequence can also converge to +∞or to −∞ (or perhaps it’s better to say “diverge to +∞”).
anconverges to +∞if the terms increase without bound, i.e. limn→∞ an= +∞if for any given
> 0 there exists an Nsuch that for all n > N, < an. An unbounded monotonic sequence
converges to infinity.
1. §Show that the sequence defined by a0= 2 and an+1 = 1/(3 −an) is positive and decreasing.
Hence it must be convergent. Find the limit.
2. (a) Show that the sequence {√2,p2√2,q2p2√2, . . . }converges. This takes two steps:
show that each term is bigger than the previous, and show that no term is bigger than
3 (by induction: show that if anis less than 3, then so is an+1 ).
(b) §Find the limit of the sequence {√2,p2√2,q2p2√2, . . . }, by finding an equation the
limit must satisfy.
3. §Determine whether the following sequences converge to a finite number, converge to ±∞,
1
