I implemented the Newton method to find the non-zero root of $f(x) = 1-bx-e^{-x}$ in Excel and I have tested it for various values of $0<b<1$. However, what I am seeing for some values of $b$ (e.g. $b=0.9$) is that the solutions oscillates around some values without converging. (My starting value is $x_0 = 1/b$.) As far as I can tell, all the requirements for the quadratic convergence of the method are satisfied, so I was excepting the solution to converge, but the results oscillate around the solution with absolute error of $err \approx 5.55E-15$. Since for doubles the machine precision is $\epsilon = 2^{-52}\approx 2.22E-16$ it seems to me that there is something else at play. However, peculiarly, $err/\epsilon = 25.0$ exactly (within Excel's precision).
So, I guess my question is: how can I determine whether a convergence has been reached within machine precision?