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MATH 138
Winter 2011
Assignment 6
Topics: Sequences, series.
Due: 11 a.m. Friday, February 18.
1. Prove the following theorem in two ways: “If
lim
n
→∞

a
n

= 0
, then
lim
n
→∞
a
n
=
0
.”
(a) Using the squeeze theorem for sequences.
Proof:
We have

a
n
 ≤
a
n
≤ 
a
n

. We have
lim
n
→∞

a
n

=

lim
n
→∞

a
n

=
0 = lim
n
→∞

a
n

so by the squeeze theorem
lim
n
→∞
a
n
= 0
.
(b) Using the deﬁnition of limits.
Proof:
Since
lim
n
→∞
a
n
= 0
, we know that
∀
± >
0
, there exists
N
such
that if
n > N
, we have
±
±

a
n

0
±
±
< ±
. That last condition is of course
equivalent to

a
n

< ±
so
lim
n
→∞
a
n
= 0
.
2. Determine whether the sequence converges or diverges. If it converges, ﬁnd
the limit.
(a)
a
n
=
(
n

1)!
(
n
+1)!
Answer:
Note that
a
n
=
1
n
(
n
+1)
. Since the sequences
{
1
n
}
and
{
1
n
+1
}
converge both to
0
, their product
a
n
also
converges
to
0
.
(b)
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This note was uploaded on 07/24/2011 for the course MATH 138 taught by Professor Anoymous during the Winter '07 term at Waterloo.
 Winter '07
 Anoymous
 Math, Squeeze Theorem

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