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For a particular problem we may have to verify the arithmetic of an infinite sequence. We have seen in the first part of this chapter how this can be done by computing the limit of the sequence. This would not be possible if the sequence had an infinite amount of repetitions. We will illustrate this by running through some simple examples.

Finding the limit of a sequence

First we will show how you can find the limit of a sequence.

We have a sequence of numbers:

…

We can notice that it is a sequence of increasing numbers and we have the following expression

…

The L in the right hand side stands for the limit of the sequence. Let’s find the limit of the sequence

S

n

=

3,

4,

5,

7,

8,

11,

12,

15,

16,

23,

24,

31,

34,

41,

44,

57,

58,

71,

74,

87,

88,

99,

100,

121,

124,

145,

148,

…

If we have a sequence of numbers s

n

1

s

n

2

3

4

5

7

8

11

12

15

16

23

24

31

34

41

44

57

58

71

74

87

88

…

and we want to find the limit (if it exists) of the sequence s

S

n

Then we need to use the Eq (8.1.3)

By simple calculations we can see that the limit of the sequence is

…

The properties of this sequence are:

– It has an infinite amount of elements

– It is monotonically increasing.

Now let’s see a sample computer program, called LIMITSUB that checks whether a given sequence has a limit or not.

Sequence x has a limit? Example

LIMITSUB. A program to find the limit of a sequence.

#!/usr/bin/per
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