Math 409-502 Harold P. Boas

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Math 409-502

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 409-502 September 13, 2004 — slide #2

Examples of error term estimation

Powers

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.

Geometric series

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 409-502 September 13, 2004 — slide #3

Examples continued

Newton’s method for approximating

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

|

<

Fibonacci fractions

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 409-502 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 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

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