Fixed Point Iteration Method

Since we have some idea of how the exponential function behaves we can try to guess the solution. We know that:

and

and

We can guess that the solution for should lie between 0.5 and 1.

However, this is a rough estimate. Surprisingly if we take an initial guess value :

and iterate as follows starting with k=0:

then x1=0.606, x2=0.545 , x3= 0.579, x4=0.5600, x5=0.571, x6=0.565, x7=0.568, x8=0.566, x9=0.567 ....

We seem to be "converging" to a solution which satisfies the equation !