determine their sum, difference, or order.

More infinite quantities
------------------------

Some infinite sequences represent finite quantities, although they are composed of an infinite multitude of fractions, for instance irrational quantities or geometrical series.

Let $e < 1$.

1.  $$\frac{14}{10} + \frac{1}{100} + \frac{4}{1000} + \frac{2}{10000} + \dots + \text{in inf.} = \sqrt 2$$

2.  $$a + ae + ae^2 + ae^3 + \dots + \text{in inf.} = \frac{a}{1-e}$$

Bolzano presents examples of infinite quantities of higher orders and infinite quantities which have any rational or irrational order. So he extends the notion of the infinite quantities.

Let $\alpha$ be any quantity then the following series are also quantities. $\alpha \cdot N$ is also a quantity. (§29).

1.  $$\alpha \cdot N = \alpha + \alpha + \alpha + \dots + \text{in inf.}$$

2.  $$N^2 = N + N + N + N + \dots + \text{in inf.}$$

3.  $$N^3 = N^2 + N^2 + N^2 + N^2 + \dots + \text{in inf.}$$

\[pomer\]

1.  Let $\alpha, \beta$ be two quantities then $$(\alpha \cdot N) : (\beta \cdot N) = \alpha : \beta.$$

2.  $N^2$ is infinitely greater than $N$, $$N^2 >> N.$$

3.  $N^3$ is infinitely greater than $N^2$, $$N^3 >> 