


Benoit Cloitre is a brilliant maths amateur when he wants. Consider the sequences _{n} and _{n} to be defined as
In general, for
Then lim_{ n} = x + By considering those two sequences, e and are mirrored! However, the speed of convergence are very different... U_{n} converges very quickly (faster and faster) while V _{n} converges in a logarthmic way... 2 ProveBenoit Cloitre first offered a direct prove, but one can prefer Gery Huvent's prove, from Lille, which allow us to find a general method to prove those sequences. 2.1 For _{n}Calculating the first few terms 0,1,1,,,, suggest that This property can be prove by induction on p, since it's true for p = 1. Suppose that v_{2p} = v_{2p+1} then and
which by induction give us Taking into account the classical equivalence (obtain with Wallis) we have
Note We can find a limited development with Stirling's formula, to get Consider the function that generates the sequence _{n} the given recurrent sequence can be written as or by differentiating, we get hence f is the solutiong to the differential equation which has the general solution taking into acount than u_{1} = 0 and u_{2} = 1 = we find Now we need to determine the development of the power serie of f so to find each term of the sequence. For this note that ( = v_{2}x
+ v_{3}x^{2} + ....is defined in 0 )
then using Cauchy's product of two series. Hence and
and more particularly
Note If we
did the same for the sequence
_{n}, the generating
function is a solution to
which satisfy with
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