MATH 8, SECTION 1  MIDTERM REVIEW NOTES
TA: PADRAIC BARTLETT
Abstract.
These are the notes from Friday, Oct. 25th’s midterm review.
The following is kind of a condensed, “CliffsNotes”style review of the course
thus far; here, we list all of the major theorems and results we’ve covered thus far,
and talk a little bit about when we would use these theorems. Basically, this is the
last four and a half weeks in one handout; if you want to see *examples* of how
these things are actually used, consult the online notes for those particular weeks!
1.
Proof Methods
(Relevant lectures:
Monday, wk. 1
,
Wednesday, wk. 1
,
Friday, wk. 1
.)
Basically, you are all – as a class –quite capable with proof methods; so there’s not a
lot to say here. However, it is worth it to mention the structure of an inductive proof
again, as it’s been a while since we’ve used induction (as opposed to direct proofs
or proofs by contradiction!), and a lot of people get tripped up on the structure of
these things:
1.1.
Proofs by Induction.
Suppose that you have a claim
P
(
n
) – a sentence like
“2
n
≥
n
”, for example. How do we prove that this kind of thing holds by induction?
Well: we generally follow the outline below:
Lemma 1.1.
P
(
n
)
holds, for all
n
≥
k
.
Proof.
Base case: we prove (by hand) that
P
(
k
) holds, for a few base cases.
Inductive step: Assuming that
P
(
m
) holds for all
k
≤
m < n
, prove that
P
(
n
)
holds.
Conclusion:
P
(
n
) holds for all
n
≥
k
.
2.
Sequences
(Relevant lectures:
Friday, wk. 2
,
Monday, wk. 3
, )
2.1.
Definitions.
A
sequence
is just an infinite collection of objects
{
a
n
}
∞
n
=1
indexed by the natural numbers.
The main property that we’ve studied about
sequences in this class is that of convergence:
Definition 2.1.
A sequence
{
a
n
}
∞
n
=1
converges to some value
λ
if, for any distance
, the
a
n
’s are eventually within
of
λ
. To put it more formally, lim
n
→∞
a
n
=
λ
iff for any distance
, there is some cutoff point
N
such that for any
n
greater than
this cutoff point,
a
n
must be within
of our limit
λ
.
In symbols:
lim
n
→∞
a
n
=
λ
iff (
∀
)(
∃
N
)(
∀
n > N
)

a
n

λ

<
.
1
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TA: PADRAIC BARTLETT
2.2.
Tools.
We have the following tools for manipulating and studying sequences:
(1)
Arithmetic and Sequences
:
•
Additivity of sequences
:
if lim
n
→∞
a
n
,
lim
n
→∞
b
n
both exist, then
lim
n
→∞
a
n
+
b
n
= (lim
n
→∞
a
n
) + (lim
n
→∞
b
n
).
•
Multiplicativity of sequences
: if lim
n
→∞
a
n
,
lim
n
→∞
b
n
both exist, then
lim
n
→∞
a
n
b
n
= (lim
n
→∞
a
n
)
·
(lim
n
→∞
b
n
).
•
Quotients of sequences
: if lim
n
→∞
a
n
,
lim
n
→∞
b
n
both exist, and
b
n
6
=
0 for all
n
, then lim
n
→∞
a
n
b
n
= (lim
n
→∞
a
n
)
/
(lim
n
→∞
b
n
).
When using these properties,
please remember
to show that both of the
limits lim
n
→∞
a
n
,
lim
n
→∞
b
n
exist before splitting them apart!
TAs will
dock you mad points for failing to do this, as it is one of the most common
ways for people to make errors in limit calculations.
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 Fall '09
 Math, Calculus, lim, Mathematical analysis, PADRAIC BARTLETT

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