Second derivitive of a dynamical system

[Daniel]

The regular formula for geometrical progressions

What do you mean by this?

That \( \sum_{k=0}^n x^k = \frac{1-x^{n+1}}{1-x} \) unless \( x^m=1 \) for some integer \( m \).

For \( x=1 \) we have \( \sum_{k=0}^n x^k = n+1 \).

Here \( x \) is the Lyponouv characteristic - the first derivitive at a fixed point. Note that the regular formula for geometrical progressions blows up for \( x=1 \) which happens at \( a=e^{\frac{1}{e}} \).