Practice Exam 1
9/22/06
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(1)
(a) Give the mathematical definition of the statement
the sequence
(
a
n
)
converges to the limit
L
.
The sequence (
a
n
) converges to the limit
L
if for any
>
0 there exists an
N >
0 such that

a
n

L

<
for every
n > N
.
(b) Explain the meaning of this definition in common English.
The definition says that given any distance, from some term onwards the
sequence will always be less than that distance away from the limit.
Alternatively, the sequence will eventually remain closer to the limit than
any specified distance.
(2)
(a) Show that 7
n
2
=
O
(
n
2

1).
We may simply show that the sequence

7
n
2
n
2

1

=
7
n
2
n
2

1
converges. We have
that
lim
n
→∞
7
n
2
n
2

1
= lim
n
→∞
7
1

1
n
2
=
7
1

0
= 7
where the secondtolast equality is due to the facts that the function
7
1

x
is continuous at
x
= 0 and that lim
n
→∞
1
n
2
= 0.
Therefore
7
n
2
n
2

1
converges and hence 7
n
2
=
O
(
n
2

1).
(b) Show that sin(
n
) =
O
(
1
2
).
For the definition of “big Oh” we must show that there are constants
C
and
N
such that

sin(
n
)
 ≤
C

1
2

for each
n
≥
N
. Since

sin(
n
)
 ≤
1 for all
n
we may just choose
C
= 2 and
use any
N
. That is to say

sin(
n
)
 ≤
1 = 2

1
2

for all
n
.
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 Fall '09
 COSTELLO
 Math, Addition, Natural number, digit numbers

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