Achill and the turtle - the problem

Before I will explain the solution of the problem, we need to have a close look on the situation, in order to shoot with some mathematical amunition. This close look will show pretty fast, that Zenon's strange idea is simply wrong.

Zenon states that Achill will never be able to catch up with the turtle. Well, we will simply try the calculate the time when Achill is at the same place where the animal is. If Zenon is right this should be an icredible big number of seconds.

The first 100 Meter passes Achill in mere 10 seconds (waving a "Hello!" at Carl Lewis). For the next ten meters this guy needs 1 second, for the next meter just 0.1 second, then 0.01 second and so on. Just like Zenon explained. If we add all these seconds we get the time when Achill is ahead with the nasty old turtle.
Well then, the calculation is quite simple. It is 10 + 1 + 0.1 + 0.01 + 0.001 + 0.0001 + ... seconds.
The "..." means, that we have to add numbers and numbers and never stop (just like Zenon said).

Zenons statement was that there is no chance for Achill. This means that the time we get by adding all the seconds is simply unmeasurable big, it is infinity. Well, is this really true? Is the result of 10 + 1 + 0.1 + 0.01 + .... really infintely big? Does it take infenitely long for Achill to catch up with the turtle?

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