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7.6 Geometric Series

7.63   Theorem ( .)If , then .

Proof:

Case 1:
[]. By the formula for factoring (3.78), we have for all and all

so

If we let in this formula, we get

Since is a null sequence, it follows from the comparison theorem for null sequences that ; i.e., .
Case 2:
[.] Let . Then , so by Case 1, . By the reciprocal theorem ; i.e., .

We have shown that the theorem holds in all cases.

7.64   Theorem (Convergence of geometric sequences.) Let . Then

Proof: The last assertion was shown in theorem 7.7, and the second statement is clear, and it is also clear that if .

Suppose that . I will show that

 (7.65)

Since is a null sequence, it follows from the comparison theorem for null sequences that is a null sequence, and then by the root theorem for null sequences (Theorem 7.19), it follows that is a null sequence.

To prove (7.65), let be a precision function for , and let . Then , so , so
. and hence , which is what we wanted to show.

7.66   Theorem (Geometric series.) Let . If , then the geometric series

converges to . If , then diverges.

Proof: We saw in theorem 3.71 that for all . If , . This sequence diverges, since it is not bounded. If , then by the previous theorem , so

Suppose now and . Then for all we have

Hence for all we have

By theorem 7.7, if and , then diverges, and hence is false for all ; i.e., diverges.

7.67   Notation. If is a sequence of digits, then we denote by . Thus

and

7.68   Example. Let be digits, and let

so informally, . Then is a convergent sequence, and

As an example, we have

7.69   Exercise. A Let

Show that converges to a rational number.

7.70   Exercise. A
a)
Let . Does converge? If it does, find in the form where .
b)
Let . Does converge? If it does, find in the form where
c)
Let . Does converge? If it does, find in the form where .

7.71   Exercise. Show that the sequences and (which are drawn above in figure b 7.1) converge, and that the limits appear to be in agreement with Figure b above.

7.72   Entertainment (Snowflakes) Let be an equilateral triangle with

Snowflakes

area , and side . Note that an equilateral triangle with side has area . Starting with , we will now construct a sequence of polygons. will have sides, all having length . We let (so has sides of length ). To construct from we attach an equilateral triangle with side of length to the middle third of each side of .

A horizontal side of might be replaced by . Each side of is replaced by 4 sides of length , so will have sides of length . The figure shows some of these polygons. I will call the polygons snowflake polygons. We have for all . The snowflake is the union of all of the sets ; i.e., a point is in if and only if it is in for some .

Find the area of (in terms of the area of ), for example

Then find the area of in terms of . Make any reasonable assumptions that you need. What is the perimeter of ?

Next: 7.7 The Translation Theorem Up: 7. Complex Sequences Previous: 7.5 Theorems About Convergent   Index