f[x_] := x + (2 - x^2)/(2*x)

f[1]

f[f[1]]

f[f[f[1]]]

N[%]

N[%, 16]

%^2

p1 = Plot[{x, f[x]}, {x, 0.1, 10}]

y[0] = 10

\(\(Do[y[n + 1] = f[y[n]], {n, 0, 10}]


    
 (*\ y[10]\ is\ a\ big\ fraction\ *) \)\)

y[10]

\(\( (*\ y[10]\ is\ very\ close\ to\ 2^0.5\ *) 

N[y[10]^2 - 2]
\)\)

\(\(Do[y[n + 1] = N[f[y[n]]], {n, 0, 10}]
 (*\ 
      now\ y[10]\ is\ a\ less\ accurate\ solution\ *) 
\)\)

N[y[10]^2 - 2]

\(\( (*graphical\ representation\ of\ the\ iterations*) 


    z[0] = y[0]\)\)

\(\(Do[z[4*l + 1] = y[l + 1]; z[4*l + 2] = y[l + 1]; 
      z[4*l + 3] = y[l + 1]; z[4*l + 4] = y[l + 1], {l, 0, 9}]
\)\)

t1 = Table[{z[2*m], z[2*m + 1]}, {m, 0, 18}]

p2 = ListPlot[t1, PlotJoined -> True, PlotRange -> All]

Show[p1, p2]