Math 409-502
Harold P. Boas boas@tamu.edu

Proving existence of lim n →
∞ a n
1. If the value of the limit is not known, we can try to use that a bounded monotonic sequence converges .
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 409-502 September 13, 2004 — slide #2
Examples of error term estimation
Powers a a n
→
1/ n
0 and na n
→ 0 when | a |
<
1;
→ 1 when 0
< a ; n 1/ n
→ 1
Strategy: use the binomial theorem to estimate the error term.
Geometric series a n
= 1 + r + r 2
+ · · · + r n
The error term is so e n e n
: = a n
−
1
1 − r
=
− r n +
1
1 − r
,
→ 0 when | r |
<
1 (by the first case above).
Math 409-502 September 13, 2004 — slide #3

Examples continued
Newton’s method for approximating
√
2
Here a n + 1
0.9.
=
1
2
( a n + a
2 n
) , and the error term satisfies a quadratic estimate | e n + 1
| ≤ | e n |
2 if | e
1
|
<
Fibonacci fractions
1
2
,
2
3
,
3
5
,
5
8
,
8
13
,
13
21
The limit of the sequence is the golden ratio
The error term satisfies | e n +
1
| ≤
1
2 n
√
5
2
−
| e
1
| if | e
1
|
<
0.2.
1
.
, . . .
Math 409-502 September 13, 2004 — slide #4
A puzzle: 1 = 0 n lim
→
∞
Z
1
( n + 1 ) x
0
= n lim
→
∞
· n dx x n +
1
¸
1
0
=
=
Z
0
Z
1
= 1 = 0
0
1 n lim
→
∞
0 dx
( n + 1 ) x n dx
What went wrong?
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 409-502 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 409-502 September 13, 2004 — slide #6