Help15:=proc(): print(`ArcA(k,N), ArcP(k,N) , AT(L), FindMachin(a,k), NorthM(N) `):end:

ArcA:=proc(k,N) local p,i:

p:=sin(Pi/k):

for i from 2 to N do
p:=sqrt((1-sqrt(1-p^2))/2):
od:

k*2^(N-1)*p:
end:



#ArcP(k,N): Archimedes' lower bounds and upper bounds using an inscribed and circumsribed polygons with k*2^N, sides. For example, to get
#the Archimedes original values of 96-gons, do Arch(3,5);
ArcP:=proc(k,N) local p,P,i:


p:=k*sin(Pi/k):

P:=k*tan(Pi/k):

for i from 2 to N do

 P:=1/2*(1/P+1/p):
 P:=1/P:

 p:=sqrt(P*p):
od:

[p,P]:

end:






#AT(L): Sum arctan(L[i],i=1..n)
AT:=proc(L)  local a,b:

if nops(L)=1 then
L[1]:
else
 a:=AT([op(1..nops(L)-1,L)]):
 b:=L[-1]:
normal((a+b)/(1-a*b)):
fi:

end:

#FindMachin(a,k): Finds the unique b such that AT([(1/a)$k,b])=1. Try FindMachin(5,4);
FindMachin:=proc(a,k) local b:
b:=solve(AT([(1/a)$k,b])=1,b):
[(1/a)$k,b]:
end:


#NorthM(N):
NorthM:=proc(N) local k:
evalf(4*add((-1)^(k-1)/(2*k-1),k=1..N)):
end: