Table[{n, Cos[2  Pi/n]}, {n, 20}]

s[n_] := Sum[1/2^k, {k, 1, n}]

s[3]

s[10]

s[50]

s[n]

N[s[3], 5]

N[s[10], 15]

N[s[50], 60]

Remove[x]

x[1] = N[1, 100]

x[n_] := x[n] = (x[n - 1] + 3/x[n - 1])/2

Table[x[n], {n, 10}]

%^2

r3 = N[Sqrt[3], 200]

Table[{n, N[x[n] - r3]}, {n, 10}]

Remove[x]

x[1] = N[1, 100]

x[n_] := x[n] = (x[n - 1] + 21/x[n - 1])/2

Table[{n, x[n]}, {n, 10}]

r21 = N[Sqrt[21], 200]

Table[{n, N[x[n] - r21]}, {n, 10}]