Suites réelles
Exercice 2
> x:=n->binomial(2*n,n);
> limit(x(n)/x(n-1),n=infinity);
> x:=n->n^n/n!;
> limit(x(n)/x(n-1),n=infinity);
> x:=n->(3*n)!/(n^(2*n)*n!);
> limit(x(n)/x(n-1),n=infinity);
Exercice 6
> limit((2^(n+1)+3^(n+1))/(2^n+3^n),n=infinity);
> limit(sum(1/2^k,k=1..n),n=infinity);
> limit(sum((-1/3)^k,k=0..n),n=infinity);
> limit(n*sin(n!-1515)/(1+n^2),n=infinity);
> limit(sum(k^2,k=1..n)/n^3,n=infinity);
> limit(sqrt(1+n+n*ln(n))-sqrt(n),n=infinity);
> limit((n+(-1)^n)/(n-(-1)^n),n=infinity);
> limit((-1)^n*(2*(-1)^n+3),n=infinity);
> limit(simplify((-1)^n*(2*(-1)^n+3)),n=infinity);
Mouais...
> limit((-1)^n*(2*(-1)^n+3/sqrt(n)),n=infinity);
> limit(simplify((-1)^n*(2*(-1)^n+3/sqrt(n))),n=infinity);
> limit(sum(cos(1/sqrt(n+k)),k=0..n)/n,n=infinity);
Je suis bluffé...
> trunc(2.8);
> limit(sum(trunc(k*x),k=1..n)/n^2,n=infinity);
> add(trunc(k*Pi),k=1..1000)/1000^2;
> evalf(%);
Ca me rappelle quelque chose...
> 2*%;
> add(trunc(k*Pi),k=1..10000)/10000^2;
> evalf(%);
> 2*%;
> evalf(Pi);
La convergence semble assez lente.
Exercice 7
> rsolve({u(0)=-1,u(n+1)=2*u(n)-3},u(n));
> limit(%,n=infinity);
> rsolve({v(0)=2,v(n+1)=v(n)/2+6},v(n));
> limit(%,n=infinity);
> rsolve({w(0)=0,w(n+1)=-w(n)+2},w(n));
> limit(%,n=infinity);
Exercice 8
> a:=n->r*n+a0;
> a(1515);
> sum(a(k-1)*a(k)*a(k+1),k=1..n);
> limit(%/a(2*n+2)^4,n=infinity);
Exercice 9
>
premiers_termes:=proc(n,u0)
local i,u,s;
u:=u0:s:=u:
for i from 1 to n do u:=1515+sqrt(u):s:=s,u od;
RETURN(s) end;
> premiers_termes(5,1024);
oops ! Il va falloir feinter...
> premiers_termes(5,1024.);
> Digits:=50;
> premiers_termes(10,1024.);
>
premiers_termes_bis:=proc(n,v0)
local i,v,s;
v:=v0:s:=v:
for i from 1 to n do v:=v^2/(1+v^2):s:=s,v od;
RETURN(s) end;
> premiers_termes_bis(5,1515);
oops again !
> Digits:=10:premiers_termes_bis(10,1515.);
Exercice 11
> asympt(n*(n^(1/n)-1),n);
> asympt(n*(n^(1/n)-1),n,1);
> asympt(sqrt(n^2+1789*n+1515)-n,n,1);
> asympt((n^1515+1789)^(1/n),n,1);
> asympt(sum(1/k^k,k=1..n),n);
Error, (in asympt) unable to compute series
> asympt(sum(k^k,k=1..n),n);
Error, (in asympt) unable to compute series
> asympt(sum(k!,k=1..n),n);
Error, (in asympt) unable to compute series
> asympt(1-cos(alpha/n),n,3);
> asympt(n*(sin(1/n)-tan(1/(2*n))),n);
> asympt(tan(sin(1/n))-sin(tan(1/n)),n);
> asympt(tan(sin(1/n))-sin(tan(1/n)),n,8);
Exercice 12
Où le taupin surpasse Maple
> sum(k^1515,k=0..n);
Warning, computation interrupted
> sum(k^20,k=0..n);
> asympt(Sum(k^1515,k=0..n),n);
> asympt(Sum(k^100,k=0..n),n);
> asympt(Sum(k^500,k=0..n),n);
Où Maple m'étonne...
> limit(sum(1/(k*(ln(k)+1024)),k=1..n),n=infinity);
> asympt(sum(1/(k*(ln(k)+1024)),k=1..n),n,1);
Exercice 13
> u:=n->n^(ln(n)^2):v:=n->(n^2)^(ln(n)):w:=n->(ln(n))^(n*ln(n)):z:=n->(n*ln(n))^n:
> limit(u(n)/v(n),n=infinity);
> limit(w(n)/u(n),n=infinity);
> limit(w(n)/z(n),n=infinity);
> limit(z(n)/u(n),n=infinity);
Ainsi, v << u << z << w
Exercice 14
> alpha:=n->(ln(ln(n)))^(-ln(n*ln(n)));beta:=n->(ln(n*ln(n)))^(-ln(ln(n)));
> limit(alpha(n)/beta(n),n=infinity);
Ainsi, alpha << beta