# 
restart;
# Use newtons method to solve x^2-4=0 with initial guess x0=1
f:=x->x^2-4;
phi:=unapply(x-f(x)/D(f)(x),x);
x0:=1;
x1:=phi(x0);
x2:=phi(x1);
x3:=phi(x2);
x4:=phi(x3);
# Find the distance between x4 and the root of x^2-4=0 which
# is closest to x4
dist:=2-x4;
evalf(dist);
# Make change of variables u=1/t in int(1/t) and then v=u^2
with(IntegrationTools);
I1:=Int(1/t,t=1..x);
I2:=Change(I1,u=1/t);
I3:=Change(I2,v=u^2);
# Note you can intepret these as log(x)=-log(1/x) and also
# as log(x)=-1/2 log (1/x^2)
# Find S2, S4, and S8 of Taylor series for exp(1/2)
S:=n->Sum((1/2)^k/k!,k=0..n);
value(S(2));
value(S(4));
value(S(8));
evalf(S(8));

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