Math 409502
Harold P. Boas [email protected]
Proving existence of
lim
n
→
∞
a n
1. If the value of the limit is not known, we can try to use that
converges
.
a bounded monotonic sequence
2. If we know or can guess that
a n
→
L
, we can try to estimate the error term
e n
:
=
a n
−
L
.
Math 409502 September 13, 2004 — slide #2
Examples of error term estimation
a a n
→
1/
n n
1/
n
0 and
→
→
1
na n
1 when 0
→
<
0 when
a
;

a

<
1;
Strategy: use the binomial theorem to estimate the error term.
a n
=
1
+
r
+
r
2
+ · · · +
r n
The error term is so
e n
→
0 when

r e n

:
=
<
a n
−
1
1
−
r
=
−
r
1
n
+
1
−
r
,
1 (by the first case above).
Math 409502 September 13, 2004 — slide #3
Examples continued
√
2
Here
0.9.
a n
+
1
=
1
2
(
a n
+
a
2
n
)
, and the error term satisfies a quadratic estimate

e n
+
1
 ≤ 
e n

2 if

e
1

<
1
2
2
3
1
2
n

e
1

3
5
5
8
The limit of the sequence is the golden ratio
The error term satisfies

e n
+
1
 ≤ if

e
1

8
13
13
21
√
5
<
−
1
2
0.2.
.
Math 409502 September 13, 2004 — slide #4
A puzzle:
1
=
0
n
lim
→
∞
Z
0
1
(
n
=
+
n
lim
→
∞
·
1
)
x n dx x n
+
1
¸
1
=
0
1
What went wrong?
=
=
=
Z
Z
0
1
1
n
lim
→
∞
(
n
0
dx
0
0
+
1
)
x n dx
The limit of the integral is not necessarily equal to the integral of the limit!
One of our goals in proving theorems about limits is to avoid mistakes like the one above.
Math 409502 September 13, 2004 — slide #5
Homework for Wednesday
1. Do Exercise 3.6/1 on page 47 and Exercise 4.4/1 on page 58.
2. Read section 5.1, pages 61–64.
Math 409502 September 13, 2004 — slide #6