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Proof: Let be a generic element of
. To simplify the
notation, I will write
Let
and let
Then for
so
It follows by the th root rule (theorem 6.48)
that
.
Hence it follows from theorem 7.6 that
|
(7.8) |
Now
Here we have used the formula for a finite geometric series. Thus, from
(7.8)
Now we want to calculate the limit appearing in the previous lemma. In order
to do this it will be convenient to prove a few general limit theorems.
Proof: Suppose for all , and .
Case 1: Suppose . Then the conclusion clearly follows.
Case 2: Suppose
. Then by the formula for a geometric series
By the sum theorem and many applications of the product theorem
we conclude that
Case 3: Suppose
. Let
. Then
,
so
by Case 2 we get
Case 4: Suppose
where
and
.
Let
. Then
Now if we could show that , it would follow from this formula
that
The next lemma shows that and completes the proof of theorem
7.10.
7.11
Lemma.
Let be a sequence of positive numbers such that , and
for all
. Then
for each in
,
.
Proof: Let be a sequence of positive numbers such that .
Let
for each in
. We want to show that
. By the formula for a finite geometric series
so
Now
Since , we have
, so by the
squeezing rule
, and hence
Proof: By lemma 7.7,
By theorem 7.10,
and putting these results together, we get
7.13
Lemma.
Let
, and let
, with . Then
Proof: If
is a partition of , let
be the partition of obtained by multiplying the points of by .
Then
|
(7.14) |
If
is a sample for , let
be the corresponding sample for .
Then
Let be a sequence of partitions of such that
, and for each
let be a sample for
. It follows from (7.14) that
.
By the area theorem for monotonic functions (theorem 7.6),
we have
Thus
Proof: The result for the case was proved in theorem 5.76.
The case is done in the following exercise.
7.16
Exercise.
A
Use the two previous lemmas to prove
theorem
7.15 for the case
.
Remark: In the proof of lemma 7.7, we did not use the assumption
until line (7.9). For equation (7.9)
becomes
Since in this case
,we conclude that
|
(7.17) |
This formula give us method of calculating
logarithms by taking square roots.
We
know
will be near to when is
large,
and
can be calculated by taking successive
square
roots. On my calculator, I pressed the following sequence of keys
and got the result . My calculator also says that
. It appears that if I know how to calculate square roots, then
I
can calculate logarithms fairly easily.
7.18
Exercise.
A
Let
be a non-negative rational number, and let
. Show that
Where in your proof do you use the fact
that
?
Next: 8. Integrable Functions
Up: 7. Still More Area
Previous: 7.1 Area Under a
  Index
Ray Mayer
2007-09-07