# Tema 4   Successions numèriques
04-4 Criteris de convergència i càlcul de límits

kill(all)$

## Exemple 04-4.1

a[n]:=sum(1/sqrt(n^2+3*k), k, 1, n);

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

x[n]:=n/sqrt(n^2)$

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

y[n]:=n/sqrt(n^2+3*n)$

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

## Exemple 04-4.2

kill(all)$

a[1]:3;         a[n]:=sqrt(2+a[n-1]);

 makelist(a[n], n, 1, 5);

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

a[1];     a[2],numer;     a[3],numer;     
a[4],numer;     a[5],numer;

Eq32:L=sqrt(2+L);

Eq32a:L^2=L+2;   
solve(Eq32a, L);

## Exemple 04-4.3

kill(all)$

a[n]:=(2*n^2+1)^(1/n);

[a[1], a[2],a[3],a[4],a[5]];

[a[1], a[2],a[3],a[4],a[5]], numer;

b[n]:=a[n+1]/a[n];

makelist(b[n], n, 1, 5);

makelist(float(b[n]), n, 1, 5);

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

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

## Exemple 04-4.4

kill(all)$

x[k]:=1/k$         X[n]:=sum(x[k], k, 1, n);

makelist(X[n], n, 1, 10);

Y[n]:=log(n);               
a[n]:=X[n]/Y[n];

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

xx[n]:=x[n+1];        YY[n]:=Y[n+1]-Y[n];           aa[n]:=xx[n]/YY[n];         
makelist(aa[k], k, 1, 4);

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