from the $(n+1)$-st term. The first $n$ terms are omitted. We designate this as $N_n$.

Bolzano shows several examples and explains how to count the infinite series.

\[n\] If we designate the number of natural numbers from the $n+1$ by $N_n$ then we obtain the equation. $$n = N - N_n.$$

Bolzano doesn’t prove this assertion, he calls it the “certain and quite unobjectionable equation from which we see how two infinite quantities $N$ and $N_n$ have a completely definite finite difference” (§29). His persuasion is based on the implicit assumption that the multitude of the terms of the series is always one and the same. Consequently, every term of the series is important. From the proof of the following theorem we see again the same principle.

\[e\] If $0 < e < 1$ then $$1 + e + e^2 + e^3 + \dots + \text{in inf.} = \frac{1}{1-e}$$

The proof begins like this

-   $S = 1 + e + e^{n-1} + e^n + e^{n+1} + \dots + \text{in inf.}$

-   $S = \ \quad \frac{1 - e^n}{1 - e} \quad + \quad e^n + e^{n+1} + \dots + \text{in inf.} \quad = \quad \frac{1 - e^n}{1 - e} 