# Tema 4   Successions numèriques
04-5  Equivalència de successions

## Exemple 04-5.1

kill(all)$

x[n]:=1/(n^2+1)$      y[n]:=sin(x[n])$      a[n]:=x[n]/(y[n])$
'lim(a[n])=limit(a[n], n, inf);

z[n]:=tan(x[n])$         b[n]:=x[n]/(z[n])$
'lim(b[n])=limit(b[n], n, inf);

## Exemple 04-5.2

kill(all)$

x[n]:=1/(n^2+1)$      y[n]:=1-cos(x[n])$
a[n]:=(y[n])/(x[n]^2/2);

'lim(a[n])=limit(a[n], n, inf);

## Exemple 04-5.3

kill(all)$

x[n]:=1/(n^2+1)$     y[n]:=exp(x[n])-1$      a[n]:=(y[n])/x[n]$
'lim(a[n])=limit(a[n], n, inf);

xx[n]:=1+x[n]$        yy[n]:=exp(x[n])$        aa[n]:=yy[n]/xx[n]$
'lim(aa[n])=limit(aa[n], n, inf);

## Exemple 04-5.4

kill(all)$

x[n]:=1/(n^2+1)$        y[n]:=log(x[n]+1)$

a[n]:=y[n]/x[n];
'lim(a[n])=limit(a[n], n, inf);

## Exemple 04-5.5

kill(all)$

x[n]:=log(y[n]);    y[n]:=n^2/(n^2+1);

z[n]:=y[n]-1;       a[n]:=x[n]/z[n];
'lim(a[n])=limit(a[n], n, inf);

## Exemple 04-5.6

kill(all)$

x[n]:=(3*n)*sum(1/k, k, 1, n);

y[n]:=(n+2)*(log(1/n))*(cos((%pi*n)/(4*n+1)));

a[n]:=x[n]/y[n]$        '(lim(a[n])) = limit(a[n], n, inf);

b[n]:=(3*n)/((n+2)*(cos((%pi*n)/(4*n+1))));
limit(b[n], n, inf);

## Equivalència de Stirling

kill(all)$

a[n]:=n!;       b[n]:=(n^n)*(exp(-n))*sqrt(2*%pi*n);

s[n]:=a[n]/b[n];
'(lim(s[n])) = limit(s[n], n, inf);