Benoit Cloitre

e and in a mirror

Benoit Cloitre is a brilliant maths amateur when he wants.

1 Two mirrored sequences

Consider the sequences _n and _n to be defined as

u1= 0, u2 = 1, un+2 = un+1 +

v1= 0, v2 = 1, vn+2 = + vn

In general, for

u1= x, u2 = y, un+2 = un+1 +

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...

Benoit 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

v2p+3	= + v2p+1 = * vp + v2p
	= vp = v2p+2

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

2.2 For _n

Consider the function that generates the sequence _n

the given recurrent sequence can be written as

or

by differentiating, we get

= 0

hence f is the solutiong to the differential equation

which has the general solution

taking into acount than u₁ = 0 and u₂ = 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

( = v₂x + v₃x² + ....is defined in 0 )
Then, we remember that if

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