Never ending

As an introduction to the topic, consider the following old problem. It is about how a population of rabbits grow.

Example 1: The growth of a population of rabbits

Assume the following:

  1. At the beginning of the first month, a pair of newly born rabbits (one female, one male) are placed into the world.
  2. Newly born pairs of rabbits mate after one month, and after another month a pair of rabbits (one female, one male) are born. From then on they give birth to a pair of rabbits every month.
  3. Rabbits do not die.

Question: how many rabbit pairs are there after one year?

An infinite sequence of numbers, such as

1,3,5,7,...1,3,5, 7,...

often has a regularity. This allows us to determine any number (or term) in the sequence. We are often interested in finding out where the sequence goes as we add more and more terms (the long-term behaviour). In the example above, it is relatively obvious: the sequence is formed by the odd numbers, hence the nn-th term of the sequence is

2n−12n-1

where n=1,2,3,...n=1,2,3, .... For example, the third term is 2⋅3−1=52\cdot 3-1=5. The sequence tends towards infinity (∞\infty), that is, the numbers get bigger and bigger if I move to the right in the sequence, and never reach an upper bound. We say that the limit of this sequence is ∞\infty.

Here is another example of a sequence:

1,0.5,0.25,0.125,...1,0.5, 0.25, 0.125, ...

This sequence is formed by dividing the previous number by 22 to get the next number in the sequence. The terms of the sequence decrease if we move to the right in the sequence, but remain positive. They must therefore approach 00. We say that the limit of this sequence is 00.

And another example: Consider the sequence

1,−1,1,−1,...1,-1,1,-1, ...

This sequence also has a regularity: the terms alternate between 11 and −1-1. This sequence does not approach anything. This sequence has no limit.

Finally the last example:

1,2,4,8,...1,2,4,8,...

This sequence is also regular: we get the new term by doubling the previous term. So the terms get bigger and bigger without a bound, and the limit is ∞\infty.

We will now discuss all this in more detail. We start with the formal definition of sequences.