# Achilles and the tortoise/Series/Exercise

The situation in the turtle paradox of Zenon of Elea is the following: a slow turtle (with speed ${\displaystyle {}v>0}$) has a starting point ${\displaystyle {}s>0}$ compared with the faster Achilles (with speed ${\displaystyle {}w>v}$ and starting point ${\displaystyle {}0}$). They start at the same time. Achilles can not catch the tortoise: when he arrives at the starting point of the tortoise ${\displaystyle {}s_{0}=s}$, the turtle is not there any more, but a little further, let's say at the point ${\displaystyle {}s_{1}=s_{0}}$. When Achilles arrives at the point ${\displaystyle {}s_{1}}$, the turtle is once again a bit further at the point ${\displaystyle {}s_{2}>s_{1}}$,

and so on. Calculate the elements of the sequence ${\displaystyle {}s_{n}}$, the associated time points ${\displaystyle {}t_{n}}$, and the respective limits. Compare these limits with the distance point where Achilles catches the turtle (you can calculate it directly from the given data).