But Euler himself (happy 306th birthday) has published himself an article entitled De Seriebus Divergentibus, where he explains that we can find results for divergent series. In this essay, he described some methods to sum divergent series, then he tried to compute one of them (1-1+2-6+24-120+...=Σ

_{n=0}

^{∞}n!), using four differents methods that leads him to the same result. Studies about divergent series were made by Ramanujan too.

But today, in traditionnal context, finding a sum for divergent series is absurd, and publications about there are rare. I have a discussion with a math teacher about such results but he says it is useless and clearly false.

I consider myself that summing divergent series is legitimate, and infinity should be reconsiderated. Here is a simple proof about divergent series, which is legitimate with p-adic numbers, but not with ordinary numbers :

S=1+2+4+8+16+...

S=1+2.1+2.2+2.4+2.8+...

S=1+2.(1+2+4+8+...

S=1+2S

S=-1

The same result can be found with the geometric series formula :

S=1+2+4+8+...=1/(1-2)=-1

In classical mathematics, this formula (S=1/(1-r)) must be used with |r|<1.

When I says 1+2+4+8+...=-1, I don't claim that when we add the powers of two, the result will magically converge to -1. I think that, seen through operations, the series have the behaviour of -1. And I also think that most divergent series could have the behaviour of a finite number or expression.

What is your opinions about divergent series and about this method of computing ?