# Question 2.1.a

# On pourrait faire directement une fonction qui calcule la somme. creer_tabp := proc (p) return [seq((p-1)!/(k*(p-k)), k=1..(p-1))]; end proc;

Zio2I0kicEc2IkYlRiVGJU83Iy1JJHNlcUclKnByb3RlY3RlZEc2JCooLUkqZmFjdG9yaWFsR0YqNiMsJjkkIiIiISIiRjJGMkkia0dGJUYzLCZGMUYyRjRGM0YzL0Y0O0YyRjBGJUYlRiU=

somme_tab := proc (tab) add(tab[i], i = 1..(nops(tab))); end proc;

Zio2I0kkdGFiRzYiRiVGJUYlLUkkYWRkRyUqcHJvdGVjdGVkRzYkJjkkNiNJImlHRiUvRi07IiIiLUklbm9wc0dGKDYjRitGJUYlRiU=

for l from 1 to 20 do somme_tab(creer_tabp(ithprime(l))); od;

IiIi

IiIj

IiM/

IiQvJg==

IighW0s+

Iio/Tm9HIw==

Ii4rO0chb0AkKQ==

IjErR0BGQ1piQg==

IjYrK0dobEM2cnRnJA==

Ij4rKzNNeiI0V0tIJDN4dkQp

IkErK1NldiNRUW93JDREbGxPbw==

IkorKytHSHQlWylbT0wxKlxcMSI9KTQlUik=

IlErKytnQFAkZmE3enVjRmdwQj51cS0jKkdxIg==

IlQrKytTVVpJKUdjNEYlMyxEP2ovTSR5QyUpW0ZH

ImVuKysrK1dEQChvU2AuRGA1WjY5J3BIMiF6ayN6Yj1NNQ==

Il9vKysrKysnNFtsbjhFJDM6KHAzUWJUIilSNVYjXFc4enkhZVk3USI=

ImlvKysrKys/Mi5vLipcRkslSD9EdUZYZD8qZV16ZiNHPmshKSl5bXF2UEBxJA==

Il1wKysrKysrI1IyQCNSZC05USxQKjNvczIhPkBAJFI2SWB0S0pBdXhVdGVuRiI=

ImdwKysrKysrZzQ7LGQzIltJYDZiTiMqSFsoXCcqKUhNdWBSN2grJD1hOi8qUmZWPF06IWV4

Il9xKysrKysrKycqb1RZZ28qeihbaSZIYyx2dXkoPm4mW1A5JCopPSFbIypHXU96NSEpKVFVXiF6eTwyaiI=

show_restes := proc (nmax) local k, p; for k from 1 to nmax do p := ithprime(k); print(irem(somme_tab(creer_tabp(p)), p)); od; end proc;

Zio2I0klbm1heEc2IjYkSSJrR0YlSSJwR0YlRiVGJT8oOCQiIiJGKzkkSSV0cnVlRyUqcHJvdGVjdGVkR0MkPjglLUkpaXRocHJpbWVHNiRGLkkoX3N5c2xpYkdGJTYjRiotSSZwcmludEdGLjYjLUklaXJlbUdGLjYkLUkqc29tbWVfdGFiR0YlNiMtSStjcmVlcl90YWJwR0YlNiNGMUYxRiVGJUYl

show_restes(20);

IiIi

IiIj

IiIh

# Question 2.1.b

somme_1surk := proc (p) return simplify(add(1/k, k=1..p-1)); end proc;

Zio2I0kicEc2IkYlRiVGJU8tSSlzaW1wbGlmeUdGJTYjLUkkYWRkRyUqcHJvdGVjdGVkRzYkKiRJImtHRiUhIiIvRi87IiIiLCY5JEYzRjBGM0YlRiVGJQ==

somme_1surk(3);

IyIiJCIiIw==

?denom

ap := p -> numer(somme_1surk(p));

Zio2I0kicEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmbnVtZXJHJSpwcm90ZWN0ZWRHNiMtSSxzb21tZV8xc3Vya0dGJTYjOSRGJUYlRiU=

ap(3);

IiIk

show_restes2 := proc (nmax) local k, p; for k from 1 to nmax do p := ithprime(k); print(irem(ap(p), p^2)); od; end proc;

show_restes2(30);

IiIi

IiIk

IiIh

# Question 3.1

int (x^n * (1-x)^m, x=0..1);

KigtSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCZJIm1HRigiIiJGLEYsRiwtRiQ2IywmSSJuR0YoRixGLEYsRiwtRiQ2IywoRitGLCIiI0YsRjBGLCEiIg==

# et on conclut avec m! = Gamma(m+1)

# Question 3.2

an := n -> (a*n + b)/(2^n * binomial(3*n, n));

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigsJiomSSJhR0YlIiIiOSRGLUYtSSJiR0YlRi1GLSkiIiNGLiEiIi1JKWJpbm9taWFsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJEYuIiIkRi5GMkYlRiVGJQ==

limit(an(n+1)/an(n), n=infinity);

IyIiIyIjRg==

# Donc converge par la r\303\250gle d'Alembert

simplify(an(n));

KigsJiomSSJhRzYiIiIiSSJuR0YmRidGJ0kiYkdGJkYnRicpIiIjLCRGKCEiIkYnLUkpYmlub21pYWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2JCwkRigiIiRGKEYt

convert(simplify(sum(an(n), n=0..infinity)), elementary);

LCYqJi1JKmh5cGVyZ2VvbUdJKF9zeXNsaWJHNiI2JTclIyIiIiIiI0YrRis3JCNGKyIiJCNGLEYvI0YsIiNGRitJImJHRidGK0YrKiZJImFHRidGKy1GJTYlNyUjRi9GLEYsRiw3JCMiIiVGLyMiIiZGL0YxRisjRisiIic=

simplify(hypergeom([1/2, 1, 1], [1/3, 2/3], 2/27));

LUkqaHlwZXJnZW9tR0koX3N5c2xpYkc2IjYlNyUjIiIiIiIjRilGKTckI0YpIiIkI0YqRi0jRioiI0Y=