**Zeno of Elea** lived in the fifth century BC. It is especially known for his mathematical paradoxes. Little is known about his life. The only sources are from the **Parmenides** of Plato and the writings of the Greek pre-Socratic to the Roman historian Diogenes Laertius ,.

## Main article: brief history of mathematics

One of the paradoxes, the most famous perhaps concerned that arrow that never reaches its target: it must indeed still browse the moist path has left. If the distance to the target is, for example 100 meters, the arrow must travel 100/2 (half) + 100/4 (a quarter) + 100/8 (the eighth) + 100/16 (the sixteenth) + …. of the distance, and so to infinity. So il put a time infinite and therefore never reach its target.

**The paradox of Achilles and the tortoise**

Another famous paradox portrays Achilles, the hero of the Trojan War, died after receiving a boom (driven by Paris) in the heel. Achilles and a tortoise will make a run. And the game is balanced, the turtle leaves with 100 meters ahead. Achilles Assume Achiles moves at 10 m / s. So Achilles takes 10 seconds to reach the starting point of the animal. During these 10 seconds, the turtle, for its part, advanced 1 meter (it moves 10 cm / s or 0.1 m / s). Achilles is going to put a tenth of a second to join the new starting point of the turtle located 1 meter in front of him. Meanwhile, the tortoise will have advanced further 10cm and etc …

Time puts Achilles to catch the runaway, also reduced as it is, is utilized by Tortoise to advance. **Achilles will never catch the tortoise! **Of course, this is wrong. But why ?

**First stage**

To browse the first 100 meters, 100 meters will Achilles / 10 meters per second, be **10 seconds** . The turtle has meanwhile traveled 10 seconds × 0.1 meters per second, or 1 small meter.

**Second step**

To cross this meter, Achilles will then need to 1/10 = **0.1 seconds** . The tortoise will therefore traveled 0.1 × 0.1 = 0.01 meter.

**The following steps**

Achilles will successively need 10 **second** s, then 0.1 **second** , then 0,001 **second, then** 0.00001 **seconds** and then 0.00000001 **second** … There is no effect due to the following stops. If we add all these fractions of time, we get a total of 10.10101010101010101 …… Seconds etc.

**A little theory**

In mathematics, the Greek symbol sigma (Σ) means « sum ». The Achilles necessary time can be written as follows:

- 100/10 + 1/10 + 0.01 / 10 + …. or, what amounts to the same:
- 10 + 10/100 + 10/10000 + 10/1000000 or with our Sigma:
- Σ 10/100
^{n}with n varying from 0 to infinity.

We recognize (for specialists) a sum of terms of a geometric sequence. A geometric sequence (for non-specialists) is a series of numbers, such as 1 to 2 – 4 – 8 – 16-32 …; or each number is derived from the previous multiplying by a certain figure (here 2). In our example, 4 = 2 × 2 and 4 × 2 = 8 and 16 = 8 × 2 …. We say that this sequence has first term 1 and as a result 2. The theory is that the sum of terms of a geometric sequence converges if:

- his first term is greater than 0;
- and its purpose is less than 1.

The general term of our suite

- Un 10/100 =
^{n}, we therefore - Un+1 = 10/100
^{n + 1}and - Un+1 = 10/100 × 10/100
^{n}, or - Un+1 = 1/100 × a

The first term of the sequence (10) is greater than 0 and the result is less than 1 (1/100). The sum of terms converges to a finite value which is reassuring for Achilles! The time it takes to reach the tortoise is not infinite. The same theory gives a way to compute the convergence value!

Σ10/100 ^{n} = 10- (1/100) ^{n + 1} / [1-1 / 100] = 10 / 0.99, because (1/100) ^{n + 1} goes to 0 as n tends towards « infinity.

**The sum of the terms of a geometric sequence**

Indeed if there is a geometric sequence Uk reason q and first term is> 0 then

- Uk = aq
^{k} - and
**Sn = a Σq**^{k}= ax [1-q^{n + 1}/ (1-q)]

**Sample the sum**

- S
_{n}= a + aq + aq aq² + … +^{n}; - we can multiply both sides by q and we obtain:
- qS
_{n}= aq + aq aq² + … +^{n}+ aq^{n + 1} - S
_{n}– q × S_{n}= a – aq^{n + 1} **S**_{n}= a (1-q^{n + 1}) / (1-q)

### Sample convergence

S _{n} = a (1-q ^{n + 1} ) / (1-q)

- if q <1, then
**Sn**tends to a / 1-q (q^{n + 1}approaches 0) - if q> 1, then (q
^{n + 1}tends towards ∞) and thus**Sn**too.

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