6.1. Image Sequences

The preceeding chapter has seen sequences as subjects of their own. Now we will use them as source of help to study functions more deeply. We take sequences running along the x-axis and then watch how they are altered by a function f. Our intention is to have some properties of f revealed by this method.

Definition: If

f : A \to ℝ

is any function and

(a_{n})

a sequence in A, the sequence

(f (a_{n}))

[6.1.1]

is called the image sequence of $(a_{n})$ with respect to f.

Whereas the original sequence $(a_{n})$ is one in the x-axis, we see the image sequence $(f (a_{n}))$ running along the y-axis. This perception is sensible as it's members are values of f.

Example: Let's calculate some image sequences with respect to the square function, to the Heaviside step function H
i

$H (x) = {\begin{array}{l} 1, falls x \geq 0 \\ 0, falls x < 0 \end{array}$
and to the reciprocal function.

$f = X^{2}$
- For $(n - 1)$ we get $(f (n - 1)) = ({(n - 1)}^{2})$ as image sequence.
- For $({(- 1)}^{n})$ we get $(f ({(- 1)}^{n})) = ? ({(- 1)}^{2 n}) = (1)$ as image sequence.
- For $(\frac{1}{n})$ we get $(f (\frac{1}{n})) = (\frac{1}{n^{2}})$ as image sequence.
$f = H$
- For $(\frac{1}{n})$ we get $(f (\frac{1}{n})) = (1)$ as image sequence.
- For $(- \frac{1}{n})$ we get $(f (- \frac{1}{n})) = ? (0)$ as image sequence.
- For $(\frac{{(- 1)}^{n}}{n})$ we get $(f (\frac{{(- 1)}^{n}}{n})) = ({\begin{array}{l} 1, if n is even \\ 0, if n is odd \end{array})$ as image sequence.
$f = \frac{1}{X}$
- For $(n)$ we get $(f (n)) = (\frac{1}{n})$ as image sequence.
- For $(\frac{1}{n^{2}})$ we get $(f (\frac{1}{n^{2}})) = ? (n^{2})$ as image sequence.

It is clear from the examples above that properties of the original sequence are not always passed on to the corresponding image sequence. E.g. bounded sequences could be turned into unbounded ones, also convergence is by far not always inherited from the original sequence. Contrariwise new properties may arise for an image sequence.

Thus we have the option to distinguish those functions that are faithful to certain properties of sequences.

6.2.