Suites réelles

Exercice 2

> x:=n->binomial(2*n,n);

x := proc (n) options operator, arrow; binomial(2*n...

> limit(x(n)/x(n-1),n=infinity);

4

> x:=n->n^n/n!;

x := proc (n) options operator, arrow; n^n/n! end p...

> limit(x(n)/x(n-1),n=infinity);

exp(1)

> x:=n->(3*n)!/(n^(2*n)*n!);

x := proc (n) options operator, arrow; (3*n)!/(n^(2...

> limit(x(n)/x(n-1),n=infinity);

27*exp(-2)

Exercice 6

> limit((2^(n+1)+3^(n+1))/(2^n+3^n),n=infinity);

3

> limit(sum(1/2^k,k=1..n),n=infinity);

1

> limit(sum((-1/3)^k,k=0..n),n=infinity);

3/4

> limit(n*sin(n!-1515)/(1+n^2),n=infinity);

0

> limit(sum(k^2,k=1..n)/n^3,n=infinity);

1/3

> limit(sqrt(1+n+n*ln(n))-sqrt(n),n=infinity);

infinity

> limit((n+(-1)^n)/(n-(-1)^n),n=infinity);

1

> limit((-1)^n*(2*(-1)^n+3),n=infinity);

-5 .. 5

> limit(simplify((-1)^n*(2*(-1)^n+3)),n=infinity);

-5 .. 5

Mouais...

> limit((-1)^n*(2*(-1)^n+3/sqrt(n)),n=infinity);

-2 .. 2

> limit(simplify((-1)^n*(2*(-1)^n+3/sqrt(n))),n=infinity);

0 .. 2

> limit(sum(cos(1/sqrt(n+k)),k=0..n)/n,n=infinity);

1

Je suis bluffé...

> trunc(2.8);

2

> limit(sum(trunc(k*x),k=1..n)/n^2,n=infinity);

limit(sum(trunc(k*x),k = 1 .. n)/n^2,n = infinity)

> add(trunc(k*Pi),k=1..1000)/1000^2;

1571863/1000000

> evalf(%);

1.571863000

Ca me rappelle quelque chose...

> 2*%;

3.143726000

> add(trunc(k*Pi),k=1..10000)/10000^2;

39272577/25000000

> evalf(%);

1.570903080

> 2*%;

3.141806160

> evalf(Pi);

3.141592654

La convergence semble assez lente.

Exercice 7

> rsolve({u(0)=-1,u(n+1)=2*u(n)-3},u(n));

-4*2^n+3

> limit(%,n=infinity);

-infinity

> rsolve({v(0)=2,v(n+1)=v(n)/2+6},v(n));

-10*(1/2)^n+12

> limit(%,n=infinity);

12

> rsolve({w(0)=0,w(n+1)=-w(n)+2},w(n));

-(-1)^n+1

> limit(%,n=infinity);

0 .. 2

Exercice 8

> a:=n->r*n+a0;

a := proc (n) options operator, arrow; r*n+a0 end p...

> a(1515);

1515*r+a0

> sum(a(k-1)*a(k)*a(k+1),k=1..n);

-1/2*a0*(n+1)*r^2+a0^3*(n+1)+3/2*r*a0^2*(n+1)^2-3/2...
-1/2*a0*(n+1)*r^2+a0^3*(n+1)+3/2*r*a0^2*(n+1)^2-3/2...

> limit(%/a(2*n+2)^4,n=infinity);

1/64*1/r

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 := proc (n, u0) local i, u, s; u :=...
premiers_termes := proc (n, u0) local i, u, s; u :=...

> premiers_termes(5,1024);

1024, 1547, 1515+sqrt(1547), 1515+sqrt(1515+sqrt(15...
1024, 1547, 1515+sqrt(1547), 1515+sqrt(1515+sqrt(15...

oops ! Il va falloir feinter...

> premiers_termes(5,1024.);

1024., 1547.000000, 1554.331921, 1554.425016, 1554....

> Digits:=50;

Digits := 50

> premiers_termes(10,1024.);

1024., 1547.000000000000000000000000000000000000000...
1024., 1547.000000000000000000000000000000000000000...
1024., 1547.000000000000000000000000000000000000000...
1024., 1547.000000000000000000000000000000000000000...
1024., 1547.000000000000000000000000000000000000000...
1024., 1547.000000000000000000000000000000000000000...

> 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 := proc (n, v0) local i, v, s; ...
premiers_termes_bis := proc (n, v0) local i, v, s; ...

> premiers_termes_bis(5,1515);

1515, 2295225/2295226, 5268057800625/10536120191701...
1515, 2295225/2295226, 5268057800625/10536120191701...
1515, 2295225/2295226, 5268057800625/10536120191701...
1515, 2295225/2295226, 5268057800625/10536120191701...
1515, 2295225/2295226, 5268057800625/10536120191701...

oops again !

> Digits:=10:premiers_termes_bis(10,1515.);

1515., .9999995643, .4999997820, .1999998605, .3846...
1515., .9999995643, .4999997820, .1999998605, .3846...

Exercice 11

> asympt(n*(n^(1/n)-1),n);

ln(n)+1/2*ln(n)^2/n+1/6*ln(n)^3/n^2+1/24*ln(n)^4/n^...

> asympt(n*(n^(1/n)-1),n,1);

ln(n)+O(1/n)

> asympt(sqrt(n^2+1789*n+1515)-n,n,1);

1789/2+O(1/n)

> asympt((n^1515+1789)^(1/n),n,1);

1+O(1/n)

> 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);

1/2*alpha^2/n^2+O(1/(n^3))

> asympt(n*(sin(1/n)-tan(1/(2*n))),n);

1/2-5/24*1/(n^2)+1/240/n^4+O(1/(n^5))

> asympt(tan(sin(1/n))-sin(tan(1/n)),n);

O(1/(n^6))

> asympt(tan(sin(1/n))-sin(tan(1/n)),n,8);

1/30*1/(n^7)+O(1/(n^8))

Exercice 12

Où le taupin surpasse Maple

> sum(k^1515,k=0..n);

Warning, computation interrupted

> sum(k^20,k=0..n);

1/21*(n+1)^21-1/2*(n+1)^20+5/3*(n+1)^19-19/2*(n+1)^...
1/21*(n+1)^21-1/2*(n+1)^20+5/3*(n+1)^19-19/2*(n+1)^...

> asympt(Sum(k^1515,k=0..n),n);

> asympt(Sum(k^100,k=0..n),n);

1/101*n^101+1/2*n^100+25/3*n^99-2695/2*n^97+298760*...

> asympt(Sum(k^500,k=0..n),n);

1/501*n^501+1/2*n^500+125/3*n^499-1035425/6*n^497+9...

Où Maple m'étonne...

> limit(sum(1/(k*(ln(k)+1024)),k=1..n),n=infinity);

infinity

> asympt(sum(1/(k*(ln(k)+1024)),k=1..n),n,1);

-ln(1024)+7169/12582912-O(-3226998787/549755813888)...

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);

infinity

> limit(w(n)/u(n),n=infinity);

infinity

> limit(w(n)/z(n),n=infinity);

infinity

> limit(z(n)/u(n),n=infinity);

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)));

alpha := proc (n) options operator, arrow; ln(ln(n)...

beta := proc (n) options operator, arrow; ln(n*ln(n...

> limit(alpha(n)/beta(n),n=infinity);

0

Ainsi, alpha << beta