p. 629

78.

Use Picard's Method to solve x - sin(x)  = 0.1.

We must find the roots of f(x) = 0.1 + sin(x) - x.

Plot[0.1+Sin[x]-x,{x,-π,2π}]

Let g(x)  =  f(x) + x =  0.1 + sin(x). We create a sequence (Subscript[x, n]) starting with a first approximation to the solution (which we get from the above graph): (Subscript[x, 0]) =  1 . The next terms in the sequence are obtained by \(TraditionalForm
      \`x\_n = \ \(g(x\_\(n - 1\)) = 0.1 + sin(x\_\(n - 1\))\)\).

g[x_]  :=  N[0.1 + Sin[x]];

\(x[0]\  = \ 1; 
x[n_]\  := \ g[x[n - 1]]\)

z  =  Table[x[n],  {n, 1, 20}]

ListPlot[z, PlotRange -> {0, 1.1}]

The sequence converges to a number close to 0.853776 which gives a good approximation to the solution of x - sin(x) =  0.1:

0.853776 - Sin[0.853776]