Page 152 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

Now let’s do another example. Let’s think about the following sequence:

n
∞
{x n } n=1 , x n = .
n +1
Exercise: Write down the first 10 terms in the sequence and determine where on the number line the
ants in the sequence are going. When you have finished, turn the page.

In this example, the terms in the sequence seem to be getting closer and closer to 1. Let’s prove that the
sequence converges to 1 using the definition of convergence of a sequence.

1. In this example, the role of L is played by 1.
2. A ghost number > 0 floats by.

3. The ghost number needs to find a giant number N ∈ N who will squash all the terms in
the sequence up to x N so that, for all n> N ,

|x n − 1| <.

4. We need to find N ∈ N such that

n +1

− 1 <, ∀n> N.

n

Let’s put 1 in disguise:
n n +1 n +1 n 1
1= , so − 1= − = .
n n n n n

So, we need to find N ∈ N such that

1

∀n> N, <.
n

Does this look familiar?

1
5. By our last example, the giant number N ∈ N can be any N ∈ N which is greater than .

Then, for all the “ants” x n with n> N ,

n
x n = ,
n +1

and as we have computed in step 4,

n +1 1
x n − 1= − 1= .
n n

152