Achilles and the tortoise/Series/Exercise

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

and so on. Calculate the elements of the sequence ${}s_{n}$, the associated time points ${}t_{n}$, 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).

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