restart;
# Using continued fractions to approximate sqrt(3);
Digits:=30;

x:=sqrt(3.0);

x1:=1/(x-1);

x2:=1/(x1-1);

x3:=1/(x2-2);

# The pattern repeats and is [1,1,2,1,2,1,...]
r1:=1;

r2:=1+1/1;

r3:=1+1/(1+1/2);

r4:=1+1/(1+1/(2+1/1));

r5:=1+1/(1+1/(2+1/(1+1/2)));

evalf([r1,r2,r3,r4,r5]);

# These fractions alternate too big and then too small
restart;
# Using Newton's method to approximate sqrt(3)
f:=x->x^2-3;

phi:=unapply(x-f(x)/D(f)(x),x);

x1:=phi(1);

x2:=phi(x1);

x3:=phi(x2);

x4:=phi(x3);

evalf([x1,x2,x3,x4]);

sqrt(3.0);

evalf([x1-sqrt(3),x2-sqrt(3),x3-sqrt(3),x4-sqrt(3)]);

# These fractions are all larger than sqrt(3)
restart;
# Using the Taylor series to approximate sqrt(3)
S:=n->Sum(binomial(alpha,k)*x^k,k=0..n);

alpha:=-1/2;

x:=-2/3;

r1:=value(S(1));

r2:=value(S(2));

r3:=value(S(3));

r4:=value(S(4));

r10:=value(S(10));

evalf([r1,r2,r3,r4,r10]);

# These fractions are all smaller than sqrt(3)